the optimal average information ratio of secret-sharing ......over all possible secret-sharing...

Designs, Codes and Cryptography manuscript No.(will be inserted by the editor)

The Optimal Average Information Ratio ofSecret-Sharing Schemes for the Access StructuresBased on Bipartite Graphs and Unicycle Graphs

Hui-Chuan Lu · Hung-Lin Fu

Received: date / Accepted: date

Abstract A secret-sharing scheme is a method of distributing a secret amonga set of participants in such a way that only qualified subsets of participantscan recover the secret and the joint share of the participants in any unqualifiedsubset is statistically independent of the secret. The collection of all qualifiedsubsets is called the access structure of the scheme. In a graph-based accessstructure, each vertex of a given graph G represents a participant and eachedge of G represents a minimal qualified subset. The average information ratioof a perfect secret-sharing scheme realizing a given access structure is the ratioof the average length of the shares given to the participants to the length ofthe secret. The infimum of the average information ratio of all possible perfectsecret-sharing schemes realizing an access structure is called the optimal aver-age information ratio of that access structure. In this paper, we derive boundson the optimal average information ratio of the access structures based ongeneral graphs and investigate the value of the ratio for bipartite graphs andunicycle graphs. We propose conditions under which the exact value of theoptimal average information ratio can be determined. Consequently, we deter-mine the exact values of the optimal average information ratio of some infiniteclasses of bipartite graphs and unicycle graphs. This extends previous results.We also provide a good bound on the optimal average information ratio forall unicycle graphs.

The work was supported in part by the Ministry of Science and Technology of Taiwan underGrants MOST 104-2115-M-239-001 and NSC 100-2115-M-009-005-MY3

Hui-Chuan Lu (corresponding author)Center for Basic Required Courses, National United University, Maioli, Taiwan 36003Tel.: +886-37-382777Fax: +886-37-382488E-mail: [email protected]; [email protected]

Hung-Lin FuDepartment of Applied Mathematics, National Chaio Tung University, Hsinchu, Taiwan30010

2 Hui-Chuan Lu, Hung-Lin Fu

Keywords Secret-sharing scheme · average information ratio · completemultipartite covering · star decomposition · maximum independent set ·bipartite graph · unicycle graph

Mathematics Subject Classification (2000) 05C70 · 94A62 · 94C15 ·94C17

1 Introduction

Motivated by the problem of secure information storage, secret-sharing schemeshave found numerous applications in cryptography and distributed comput-ing. A secret-sharing scheme involves a dealer who has a secret, a finite set Pof participants and a collection Γ of subsets of P called the access structure.Each subset in Γ is a qualified subset. A secret-sharing scheme is a method bywhich the dealer distributes a secret among the participants in P such thatonly the participants in a qualified subset can recover the secret and the jointshare of the participants in any unqualified subset is statistically independentof the secret. An access structure is naturally required to be monotone, that is,any subset of P containing a qualified subset must also be qualified. Therefore,an access structure Γ is completely determined by the family of all its minimalsubsets which is called the basis of Γ .

Shamir [20] and Blakley [3] independently introduced the first kind ofsecret-sharing schemes called the (t, n)-threshold schemes in 1979. In such ascheme, the basis of the access structure consists of all t-subsets of the partici-pant set of size n. Their work has raised a great deal of interest in the researchof many aspects of secret-sharing problems. Secret-sharing schemes for variousaccess structures as well as many modified versions with additional capacitieswere widely studied. The information ratio and the average information ratioof secret-sharing schemes have been the main subjects of discussion. The in-formation ratio of a secret-sharing scheme is the ratio of the maximum length(in bits) of the share given to a participant to the length of the secret, whilethe average information ratio of a secret-sharing scheme specifies the ratio ofthe average length of the shares given to the participants to the length of thesecret. These ratios represent the maximum and the average number of bits aparticipant has to remember for each bit of the secret respectively. As opposedto them, some literatures use information rate and average information ratewhich are exactly the reciprocal of the information ratio and the average in-formation ratio respectively. For lower storage and communication complexity,these ratios are expected to be as low as possible. The problem of construct-ing secret-sharing schemes with the lowest these ratios arose naturally. Givenan access structure Γ , the infimum of the (average) information ratio of allpossible secret-sharing schemes realizing this access structure Γ is referred toas the optimal (average) information ratio of Γ .

In this paper, we consider graph-based access structures. Given a simplegraph G, we let each vertex of G represents a participant and each edge of Grepresent a minimum qualified subset. That is, in the access structure based on

Optimal Average Information Ratio of Bipartite Graphs and Unicycle Graphs 3

G, the vertex set V (G) of G is the set of participants and the edge set E(G)of G is the basis of this access structure. A secret-sharing scheme Σ for theaccess structure based on G is a collection of random variables ζS and ζv forv ∈ V (G) with a joint distribution such that

(i) ζS is the secret and ζv is the share of v;(ii) if uv ∈ E(G), then ζu and ζv together determine the value of ζS ;(iii) if A ⊆ V (G) is an independent set in G, then ζS and the collection {ζv|v ∈

A} are statistically independent.

Recall that given a discrete random variable X with possible values {x1,x2, . . ., xn} and a probability distribution {p(xi)}ni=1, the Shannon entropyof X is defined as H(X) = −

∑ni=1 p(xi) log p(xi) which is a measure of

the average uncertainty associated with X. This value reflects the averagenumber of bits needed to represent the element in X faithfully [15]. UsingShannon entropy, the information ratio of the scheme Σ can be defined asRΣ = maxv∈V (G){H(ζv)/H(ζS)} and the average information ratio of Σ isARΣ = (

∑v∈V (G)H(ζv))/(|V (G)|H(ζS)). For simplicity, with the same sym-

bol G, we will denote both the graph as well as the access structure basedon it. For instance, “a secret-sharing scheme on G” refers to “a secret-sharingscheme for the access structure based on G”. Furthermore, the optimal infor-mation ratio R(G) of G and the optimal average information ratio AR(G) ofG are the infimum of the information ratio RΣ and the average informationratio ARΣ over all possible secret-sharing schemes Σ on G respectively. It iswell known that R(G) ≥ AR(G) ≥ 1 [10] and that R(G) = 1 if and only ifAR(G) = 1. A secret-sharing scheme Σ with the optimal ratio RΣ = 1 orARΣ = 1 is called ideal. An access structure G is ideal if there exists an idealsecret-sharing scheme on it.

Due to the difficulty of determining the exact values of R(G) and AR(G),most known results give bounds on them [4–9], [11,12,16,18], [21–23]. Theexact values of the optimal average information ratio of most graphs of orderno more than five and the optimal information ratio of most graphs of orderno more than six have been determined [7,16,18]. Before 2007, apart from aspecially defined class of graphs [4], the paths and cycles were the only infiniteclasses of graphs which have known optimal information ratio and optimalaverage information ratio. Csirmaz and Tardos’s [14] excellent work appearedin 2007. They determined the exact value of the optimal information ratio of alltrees. In 2009, Csirmaz and Ligeti [13] gave the best reult so far on the optimalinformation ratio of graphs by showing that R(G) = 2 − 1/d, where d is themaximum degree of G, for any graph G satisfying the following conditions: (i)every vertex has at most one neighbor of degree one; (ii) vertices of degree atleast three are not connected by an edge; (iii) the girth of G is at least six. In2012, Lu and Fu [19] went on settling the exact values of the optimal averageinformation ratio of all trees. Recently Beimel et al. [2] investigated the totalshare size of secret-sharing schemes realizing very dense graphs and showedexcellent results. In this paper, we deal with the optimal average informationratio of bipartite graphs and unicycle graphs.


This paper is organized as follows. In Section 2, we recall basic definitionsand restate some known results. In Section 3, we derive bounds on the optimalaverage information ratio for general graphs. In Section 4, we investigate theproblem for bipartite graphs and propose conditions under which the value ofthe optimal average information ratio can be determined. As a consequence,we determine the exact value of this ratio for some infinite classes of bipartitegraphs. This extends the result in [19]. In Section 5, we determine the valueof the optimal average information ratio for some unicycle graphs and give agood bound on it for all unicycle graphs. A concluding remark will be givenin the final section.

2 Preliminaries

In this section, we introduce some basic notions and important results for ourdiscussion in the following sections. All graphs considered in this paper aresimple graphs without loops and isolated vertices. The vertex number |V (G)|of a graph G is called the order of G and the edge number |E(G)| of G is thesize of it.

Birckell and Davenport [8] have given a complete characterization of idealgraph-based access structures.

Theorem 1 ([8]) Suppose that G is a connected graph. Then R(G) = AR(G) =1 if and only if G is a complete multipartite graph.

For a non-ideal access structure G, we consider upper bounds and lowerbounds on AR(G). The average information ratio of any secret-sharing schemeon G makes a natural upper bound on AR(G). Stinson’s decomposition con-struction [23] enables us to build up a secret-sharing scheme on a graph via itscomplete multipartite covering. A complete multipartite covering of a graph Gis a collection of complete multipartite subgraphs Π = {G1, G2, . . . , Gl} of Gsuch that each edge of G appears in at least one subgraph in this collection.The sum nΠ =

∑li=1 |V (Gi)| is called the vertex-number sum of Π.

Theorem 2 ([23]) Suppose that Π = {G1, G2, . . . , Gl} is a complete multi-partite covering of a graph G of order n. Then there exists a secret-sharingscheme Σ on G with average information ratio ARΣ = nΠ/n.

This theorem suggests that a complete multipartite covering of a graphwith smaller vertex-number sum leads to a secret-sharing scheme on thatgraph with lower average information ratio. A complete multipartite cover-ing with the least vertex-number sum is sald to be optimal. A good completemultipartite covering is our main tool to establish upper bounds on AR(G).In the case that every subgraph in a complete multipartite covering is a star,this covering is also called a star covering. If each edge of G appears in exactlyone subgraph in a covering, then this covering is in fact a decomposition. Inthe discussion of seeking star coverings with the least vertex-number sum, it


is sufficient to only consider star decompositions. A star decomposition worksespecially well for graphs of larger girth.

For the derivation of lower bounds on AR(G), we use a method based oninformation theoretic approach [4–6], [10–14], [16]. Let Σ be a secret-sharingscheme in which ξS is the random variable of the secret and each ξv is therandom variable of the share of v, v ∈ V (G). Define a real-valued function fas f(A) = H({ξv : v ∈ A})/H(ξs) for each subset A ⊆ V (G), where H is theShannon entropy. Then, ARΣ = 1

n

∑v∈V (G) f(v), where n is the order of G.

Csirmaz et al. [13] defined a core of G as a set of vertices V0 ⊆ V (G)satisfying that (i) V0 induces a connected subgraph in G; (ii) each vertexv ∈ V0 has a designated outside neighbor which is defined as a neighbor v ofv that is outside V0 and is not adjacent to any other vertex in V0, and (iii){v|v ∈ V0} is an independent set in G. Using the idea of a core, they show thefollowing theorem.

Theorem 3 ([14]) Let V0 be a core of a graph G. If f is defined as above,then

∑v∈V0

f(v) ≥ 2|V0| − 1.

Based on the facts introduced in this section, we shall derive lower boundsand upper bounds on AR(G) in the following section.

3 Lower bound and upper bound on AR(G)

For deriving lower bounds on AR(G), we define a core cluster of G of size k asa partition C = {V1, V2, . . . , Vk} of V (G) such that each Vi, i ∈ {1, 2, . . . , k},is a core of G. We denote the minimum size of a core cluster of G as c∗(G).A core cluster of size c∗(G) is called an optimal core cluster of G. With thisdefinition, we have a lower bound on AR(G) for general graphs.

Theorem 4 If G is a graph of order n, then AR(G) ≥ 2− c∗(G)n .

Proof Let C = {V1, V2, . . . , Vk} be a core cluster of G. If Σ is a secret-sharing

scheme on G, then we have∑v∈V (G) f(v) =

∑ki=1

∑v∈Vi f(v) ≥

∑ki=1(2|Vi|−

1) = 2|V (G)| − k, where f is the function defined in Section 2. Thus ARΣ ≥1n (2n − k) = 2 − k

n . Since this inequality holds for any secret-sharing schemeon G, we have the result. ut

This proof is quite similar to the one used to show the similar result fortrees in [19]. Note that our definition of a core cluster is slightly differentfrom the one used in [19] in which a core cluster is a partition of the set ofvertices whose degree are at least two instead of a partition of V (G). Usingour definition, the main result in [19] can be written in a neater way as follows.

Theorem 5 ([19]) If T is a nontrivial tree of order n, then AR(T ) = 2− c∗(T )n .

Combining Theorem 2 and Theorem 4, we have the following bound onAR(G) for general graphs.


Theorem 6 If G is a graph of order n and Π is a complete multipartite

covering of G with vertex-number sum nΠ , then 2− c∗(G)n ≤ AR(G) ≤ nΠ

n .

If there exists a complete multipartite covering (decomposition) Π of Gwith vertex-number sum nΠ and a core cluster C of G of size k such that2|V (G)| − k = nΠ , then we called the graph G realizable and therefore

AR(G) = 2− c∗(G)n .

Next, we consider upper bounds on AR(G) derived from star decomposi-tions ofG. Any star decompositionΠ = {S1, S2, . . . , Sl} of a graphG naturallyinduces an orientation on G in the following way. For i ∈ {1, 2, · · · , l}, we di-rect each edge of the star Si from the center toward the leaf. Let us denote theresulting directed graph as GΠ and the number of the sink vertices in GΠ assΠ . Since the set of sink vertices in GΠ is an independent set in G, we havesΠ ≤ α(G). Let us introduce the following lower bound on the vertex-numbersum of a star decomposition (covering) of a general graph.

Lemma 1 The vertex-number sum of any star covering of a graph G is atleast |V (G)|+ |E(G)| − α(G).

Proof Considering star decompositions is sufficient. Let Π = {S1, S2, . . . , Sl}be a star decomposition of G and GΠ be the directed graph induced by Π.Since we may assume that each vertex is the center of at most one star in thisdecomposition, we have l = |V (G)| − sΠ . Now, the vertex-number sum nΠ ofΠ can be evaluated as follows.

nΠ =

l∑i=1

|V (Si)| = l +

l∑i=1

|{leaves of Si}| = (|V (G)| − sπ) + |E(G)|

≥ |V (G)|+ |E(G)| − α(G)

utOn the other hand, given an independent set L in G, we are able to con-

struct a star decomposition Π such that the set of sink vertices in the directedgraph GΠ induced by Π is exactly the given independent set L.

Proposition 1 For any independent set L in G, there exists a star decom-position Π of G such that L is the set of all sink vertices in GΠ and nΠ =|V (G)|+ |E(G)| − |L|.

Proof For the given independent set L, we define an orientation on G. First,since each edge is incident to at most one vertex in L, we can direct all incidentedges of any v ∈ L toward v. Second, for an endvertex u of a directed edgeuw, if there is any undirected edge incident to u, then u must be the tail ofthe edge uw. Let us direct all undirected edges incident to u toward u. Thismakes u a nonsink vertex. As G is finite, all edges will eventually be directedby repeatedly doing the second step. Note that in the resulting directed graphG, the set of all sink vertices is exactly L. Besides, if we let Su, for eachu ∈ V (G) \L, be the star whose center is u and whose leaves are the heads of


the edges which share the same tail u. Then Π = {Su|u ∈ V (G) \ L} is therequired star decomposition with nΠ = |V (G)|+ |E(G)|− |L| and the directedgraph G is exactly GΠ . ut

Therefore, the star decomposition is optimal when the given independentset is maximum.

Corollary 7 Any graph G has an optimal star decomposition whose vertex-number sum equals |V (G)|+ |E(G)| − α(G).

Using this result, the bound on AR(G) in Theorem 6 can be written asfollows.

Theorem 8 If G is a graph of order n and size m, then 2− c∗(G)n ≤ AR(G) ≤

n+m−α(G)n .

In the directed graph GΠ induced by some star decomposition Π of G, thein-degree of v is denoted as d−GΠ

(v). The subgraph of GΠ induced by a subsetV ′ of the vertex set of G is written as GΠ [V ′]. Let C = {V1, V2, · · · , Vk} bea core cluster of G, we use m−Vi to denote the number of edges of GΠ whoseheads are in Vi and whose tails are not in Vi, for each i = 1, 2, · · · , k. Inaddition, the minimum length of a cycle in G is referred to as the girth of Gand is denoted as girth(G). In order to have the upper bound and the lowerbound on AR(G) in Theorem 8 meet, we investigate the relationship betweenan optimal core cluster and an optimal star decomposition of G in the nexttheorem.

Theorem 9 If girth(G) ≥ 5, then G is realizable if and only if there exists astar decomposition Π and a core cluster C of G such that each core Vi in Csatisfies the following conditions:

(i) Vi induces a tree in G;(ii) d−GΠ [Vi]

(v) = dG(v)−1 for each sink vertex v ∈ Vi of GΠ and d−GΠ [Vi](u) =

d−GΠ(u) for each nonsink vertex u ∈ Vi of GΠ .

Proof Let Π be a complete multipartite covering and C = {V1, V2, . . . , Vk} bea core cluster of G. From previous discussion, we know that 2|V (G)| − k ≤|V (G)|AR(G) ≤ nΠ . Since girth(G) ≥ 5, we may assume that Π is a stardecomposition and then we have

k∑i=1

(2|Vi| − 1) ≤ |V (G)|+ |E(G)| − sΠ

=

k∑i=1

|Vi|+k∑i=1

(|E(GΠ [Vi])|+m−Vi

)−

k∑i=1

|{sinks of GΠ in Vi}|

=

k∑i=1

(|Vi|+ |E(GΠ [Vi])|+m−Vi − |{sinks of GΠ in Vi}|

)


Note that 2|Vi| − 1 ≤ |Vi| + |E(GΠ [Vi])| and the equality holds if and onlyif Vi induces a tree in G. In addition, since Vi is a core, each sink vertexof GΠ has a designated outside neighbor outside Vi. This leads to the factm−Vi − |{sinks of GΠ in Vi}| ≥ 0, and the equality holds if and only if eachsink vertex of GΠ in Vi has exactly one neighbor outside Vi and each nonsinkvertex of GΠ in Vi has no incident edges going from outside GΠ [Vi] into it.Therefore, 2|Vi| − 1 = |Vi|+ |E(GΠ [Vi])|+m−Vi − |{sinks of GΠ in Vi}| if and

only if (i) Vi induces a tree in G and (ii) d−GΠ [Vi](v) = dG(v)− 1 for each sink

vertex v ∈ Vi of GΠ and d−GΠ [Vi](u) = d−GΠ

(u) for each nonsink vertex u ∈ Viof GΠ . In this case, 2|V (G)| − k = nΠ and G is realizable. ut

For graphs with smaller girth, a complete multipartite covering with theleast vertex-number sum may not be a star covering. Therefore, only the suf-ficiency is true.

Corollary 10 If there exists a star decomposition Π and a core cluster Cof G satisfying the conditions stated in Theorem 9, then G is realizable and

AR(G) = |V (G)|+|E(G)|−α(G)|V (G)| .

The discussion in the proof of Theorem 9 also leads to a bound on AR(G)as follows.

Corollary 11 If Π is an optimal star decomposition of G and C = {V1, · · · , Vk}is a core cluster of G, then

|V (G)|+ |E(G)| − α(G)− β(Π,C )

|V (G)|≤ AR(G) ≤ |V (G)|+ |E(G)| − α(G)

|V (G)|,

where β(Π,C ) =∑ki=1

(|E(GΠ [Vi])| − |Vi|+ 1 +m−Vi − |{sinks of GΠ in Vi}|

).

Note that condition (ii) in Theorem 9 requires that there is no edge goingfrom outside GΠ [Vi] into a nonsink vertex of GΠ in Vi. We have the followingobservations.

Corollary 12 If G is realizable with girth(G) ≥ 5 and both endvertices of anedge uv ∈ E(G) are centers of some stars in an optimal star decomposition ofG, then u and v must be in the same core in any optimal core cluster of G.

Corollary 13 If girth(G) ≥ 5 and there exists a maximum independent set Lin G such that G− L contains cycles, then G is not realizable.

Proof Consider the optimal star decomposition Π from Corollary 7. Since theendvertices of any edge in G − L are both centers of some stars in Π, theymust be in the same core in any optimal core cluster of G. The vertices of anycomponent in G − L are in the same core in any optimal core cluster of G.Therefore, G is not realizable because there is a core inducing cycles. ut


4 Bipartite Graphs

Now, let us consider bipartite graphs. Let G = (X,Y ) be a bipartite graphand L be a maximum independent set of G, then K = V (G)\L is a minimumvertex cover. By Konig Theorem, there exists a set of edges E(L) = E1 ∪ E2

where E1 is between L ∩X and K ∩ Y , and E2 is between L ∩ Y and K ∩Xsuch that each vertex in L∩X is on exactly one edge from E1 and each vertexin K ∩ Y is on some edge from E1, and each vertex in L ∩ Y is on exactlyone edge from E2 and each vertex in K ∩X is on some edge from E2. We callthe set E(L) a fan associated with the maximum independent set L. In thenext lemma, we will show that if the vertex set of each component in G−E(L)

is a core of G, then the core cluster {V (H)|H is a component in G − E(L)}naturally satisfies some properties required for G being realizable.

Lemma 2 Let E(L) be a fan associated with some maximum independent setL in a bipartite graph G and H be a component in G − E(L). Let Π be anoptimal star decomposition induced by L from Corollary 7. If V (H) is a coreof G, then the following statements are true.

(i) The designated outside neighbor of each v ∈ L ∩ V (H) must be the uniqueneighbor v of v with vv ∈ E(L) and the one of each u ∈ V (H)\L is aneighbor u ∈ L of u with uu ∈ E(L);

(ii) d−GΠ [V (H)](v) = dG(v) − 1 for each sink vertex v ∈ V (H) of GΠ and

d−GΠ [V (H)](u) = d−GΠ(u) for each nonsink vertex u ∈ V (H) of GΠ ;

(iii) m−V (H) = |{sink vertices of GΠ in V (H)}|.

Proof Since V (H) is a core, each vertex v in V (H) has a designated outsideneighbor which is not in V (H). On the other hand, the only neighbors of vthat are possibly outside H are the other endvertices of the incident edgesof v in E(L) and each vertex in L ∩ V (H) has at most one such neighbor.

Therefore statement (i) is obviously true and d−GΠ [V (H)](v) = dG(v) − 1 for

each sink vertex v ∈ V (H) of GΠ . According to the orientation induced byΠ, any edge whose head is u ∈ V (H)\L connects u to a vertex u′ 6∈ L andtherefore uu′ 6∈ E(L). Hence d−GΠ [V (H)](u) = d−GΠ

(u) for each nonsink vertex

u ∈ V (H) of GΠ and m−V (H) = |{sink vertices of GΠ in V (H)}|. utFrom this discussion and Corollary 11, we have the following bound on

AR(G) for bipartite graphs.

Corollary 14 If a bipartite graph G of size m and order n has a fan E(L)

associated with some maximum independent set L such that the vertex set ofeach component in G− E(L) is a core of G, then

n+m− α(G)−∑ki=1(|E(G[Vi])| − |Vi|+ 1)

n≤ AR(G) ≤ n+m− α(G)

n,

where Vi = V (Hi) and the Hi’s ,i = 1, · · · , k, are the components in G−E(L).


In this situation, if the vertex set of each component induces no cycles inG, then G is realizable.

Corollary 15 If a bipartite graph G has a fan E(L) associated with some max-imum independent set L such that the vertex set of each component in G−E(L)

is a core of G and induces no cycles in G, then AR(G) = |V (G)|+|E(G)|−α(G)|V (G)| .

Let H be a subgraph of G. For each u ∈ V (H), we let eu be an edge of Gwhich joins u to a vertex not in V (H). If we denote the set {eu|u ∈ V (H)}as E′, then the extended graph H + E′ of H is defined as the graph obtainedby adding to H all edges in E′. That is, V (H + E′) = V (H) ∪ {w|uw ∈E′ where u ∈ V (H) and w 6∈ V (H)} and E(H + E′) = E(H) ∪ E′.

Theorem 16 Suppose that E(L) is a fan associated with some maximum in-dependent set L in a bipartite graph G. For each component H in G − E(L),

let ev be an incident edge of v ∈ V (H) in E(L) and E+H = {ev|v ∈ V (H)}. If

the vertex set of the extended graph H+E+H of each component H in G−E(L)

induces no cycles in G, then AR(G) = |V (G)|+|E(G)|−α(G)|V (G)| .

Proof By Corollary 15, it suffices to show that the vetex set of each componentin G − E(L) is a core of G. Since any component H induces no cycles in G,the endvertices of each edge in E(L) lie in different components in G − E(L).For each vertex v ∈ V (H), we choose the designated outside neighbor of v asthe endvertex v of vv ∈ E+

H . Since H+E+H induces no cycles, any two distinct

vertices of H can not share the same designated outside neighbor and thedesignated outside neighbors of any two distinct vetices of H are of distanceat least two. We conclude that the set V (H) is a core of G. ut

It seems very unlikely to give a complete characterization of the graphswhich satisfy the criterion in Theorem 16. In what follows, we propose someinfinite classes of such graphs. Recall that the maximum distance of two ver-tices in a graph H is referred to as the diameter of H and is written asdiam(H).

Theorem 17 If a bipartite graph G has a fan E(L) associated with some max-imum independent set L such that diam(H) ≤ girth(G)−4 for any component

H in G− E(L), then AR(G) = |V (G)|+|E(G)|−α(G)|V (G)| .

Corollary 18 For any nontrivial tree T of order n, AR(T ) = 2 − (α(T )+1)n

and c∗(T ) = α(T ) + 1.

Proof The criterion in Theorem 17 is satisfied by any fan of a tree. Thereforewe have the value of AR(T ) as

AR(T ) =n+ |E(T )| − α(T )

n=n+ (n− 1)− α(G)

n= 2− (α(T ) + 1)

n

and c∗(T ) = α(T ) + 1. ut


The first part of Corollary 18 is equivalent to the main result in [19] whichis stated in Theorem 5 using our notation and definition. Our approach inthis paper gives a more clear description of the value c∗(T ). To introduceanother class of realizable bipartite graphs, we define the distance d(e1, e2) oftwo distinct edges e1 = u1u2 and e2 = v1v2 as d(e1, e2) = min{d(ui, vj)|i, j ∈{1, 2}}.

Theorem 19 If a bipartite graph G has a fan E(L) associated with some maxi-mum independent set L such that each cycle of G contains two edges from E(L)

which are of distance at least two, then AR(G) = |V (G)|+|E(G)|−α(G)|V (G)| .

Proof For each component H in G−E(L), we consider any cycle C = (u1, u2,· · · , uk) of G with u1 ∈ V (H). Let a be the smallest index in {1, 2, · · · , k} suchthat uaua+1 is in E(L), then 1 ≤ a ≤ k− 3. According to the assumption, anypath from u1 to ui, i ∈ {a + 1, a + 2, a + 3}, not passing the edge uaua+1hasanother edge in E(L). This edge is not incident to ua+2 and is removed from Gwhile constructing G−E(L). We therefore have that ua+1, ua+2 and ua+3 arenot vertices of H and ua+2 does not belong to any extended graph of H. Sincethis is true for any cycle containing a vertex of H, we conclude that the vertexset of any extended graph of H induces no cycles in G. The result follows byTheorem 16. ut

A k-vertex of G is a vertex whose degree is k. Another infinite class ofrealizable bipartite graphs is introduced in the next corollary.

Corollary 20 Let G be a bipartite graph. If every cycle of G contains two

2-vertices of distance at least four, then AR(G) = |V (G)|+|E(G)|−α(G)|V (G)| .

5 Unicycle Graphs

LetG be a unicycle graph. We denote the unique cycle ofG as CG = (v1, v2, · · · ,vk). For simplicity, in the discussion in this section, vi always represents vt,where t ≡ i (mod k) and 1 ≤ t ≤ k . G is called an even unicycle graph if k iseven and an odd unicycle graph if k is odd. Removing the edges of the cycleCG from G results in k trees, among which the tree containing vi is written asTi. In addition, if there exists a maximum independent set in Ti which doesnot contain vi, then Ti is called a Type I subtree of G. Otherwise, Ti is ofType II. A Type II subtree may be trivial. For unicycle graphs, the bounds onAR(G) can be written from Theorem 8 as follows.

Theorem 21 If G is a unicycle graph of order n, then 2− c∗(G)n ≤ AR(G) ≤

2− α(G)n .

Corollary 22 If G is a unicycle graph and girth(G) ≥ 5, then G is realizableif and only if α(G) = c∗(G).

Let us introduce some properties of Type I and Type II subtrees of G first.


Lemma 3 Let G be a unicycle graph with CG = (v1, v2, · · · , vk). If Ti is aType I subtree of G and u is vi−1 or vi+1, then

(i) α(Ti + u) = α(Ti) + 1 and c∗(Ti + u) = α(Ti) + 2;(ii) there is an optimal core cluster C ′ = {V ′1 , V ′2 , · · · , V ′α(Ti)+2} of Ti + u with

V ′1 = {u} such that the designated outside neighbor of vi is not u.

Proof For any independent set L′ in Ti + u, L′\{u} is independent in Ti. Wehave α(Ti + u) − 1 ≤ |L′\{u}| ≤ α(Ti), or α(Ti + u) ≤ α(Ti) + 1. On theother hand, since Ti is of Type I, there is a maximum independent set L in Tiwhich does not contain vi, then L∪{u} is an independent set in Ti+u. Hence,α(Ti+u) = α(Ti)+1 and the set L∪{u} is a maximum independent set in Ti+u.The maximum independent set L in Ti must contain at least one neighbor w ofvi in Ti such that viw is in a fan of Ti associated with L. The designated outsideneighbor of vi in the core in the optimal core cluster C of Ti constructed fromthe fan can be chosen as w. Now, by choosing the designated outside neighborof u as vi, C∪{{u}} is a core cluster of Ti+u. Since Ti+u is a nontrivial tree, weknow from Corollary 18 that c∗(Ti+u) = α(Ti+u)+1 = α(Ti)+2 = c∗(Ti)+1.Thus C ∪{{u}} is an optimal core cluster of Ti +u satisfies condition (ii). ut

Lemma 4 Let G be a unicycle graph with CG = (v1, v2, · · · , vk). If Ti is aType II subtree of G and u is vi−1 or vi+1, then

(i) α(Ti + u) = α(Ti) and c∗(Ti + u) = α(Ti) + 1;(ii) there is an optimal core cluster C ′ = {V ′1 , V ′2 , · · · , V ′α(Ti)+1} of Ti + u with

V ′1 = {u} such that vi and u are the designated outside neighbors of eachother.

Proof Since L′\{u} is independent in Ti for any independent set L′ in Ti+u, wehave α(Ti+u)− 1 ≤ |L′\{u}| ≤ α(Ti), that is, α(Ti+u) ≤ α(Ti) + 1. Supposethat α(Ti+u) = α(Ti)+1, than there exists a maximum independent set L′ inTi+u containing the vertex u. In this case, L′\{u} is a maximum independentset in Ti of size α(Ti) not containing the vertex vi. This contradicts to thefact that Ti is a Type II subtree of G. Therefore we have α(Ti + u) = α(Ti).In addition, c∗(Ti + u) = α(Ti + u) + 1 = α(Ti) + 1. Since any fan of Ti + uassociated with a maximum independent set contains the edge uvi, the optimalcore cluster of Ti + u constructed from the fan satisfies condition (ii). ut

For a unicycle graph G, we define a suitable maximum independent set inG as a maximum independent set L in G satisfying the following requirements.

(a) L contains the least number of vertices of CG = (v1, v2, · · · , vk) among allmaximum independent sets in G.

(b) For vj 6∈ L, if V (Tj) ∩ L is not a maximum independent set in Tj , then(V (Tj) ∩ L) ∪ {vj} is, j ∈ {1, 2, · · · , k}.

(c) If vi ∈ L, then (L\{vi})∪{vi−1} is not independent in G, i ∈ {1, 2, · · · , k}.

Proposition 2 If G is an odd unicycle graph with at least one Type I subtreeor an even unicycle graph, then G has a suitable maximum independent set.


Proof Let L be a maximum independent set containing the least number ofvertices of CG among all maximum independent sets in G. We adjust it tomake a suitable one. For vj 6∈ L, if V (Tj) ∩ L is not a maximum independentset in Tj , then there is a maximum independent set LTj in Tj containing vjsuch that |LTj | > |V (Tj)∩L|. Now, replacing V (Tj)∩L with LTj \ {vj} in L,we have a maximum independent set satisfying the second requirement in thedefinition. Next, for each i = 1, 2, · · · , we successively check that whether thesituation ”vi ∈ L and (L \ {vi}) ∪ {vi−1} is a maximum independent set inG” occurs or not. If so, we replace vi with vi−1 in L. We repeat this checkingand replacing process for each i = 1, 2, · · · until no vertex in L ∩ V (CG) canbe shifted forward. This process will eventually stop if G is an odd unicyclegraph with at least one Type I subtree or an even unicycle graph. ut

One can see from the proof that if G is an odd unicycle graph without TypeI subtrees, then a maximum independent set satisfying the first and secondrequirements does exist. It is the third requirement that can not be madebecause the process of shifting forward vertices in L∩V (CG) may not stop. Wehave the following two observations about a suitable maximum independentset L.

Lemma 5 If G is an odd unicycle graph with at least one Type I subtree oran even unicycle graph whose cycle is CG = (v1, v2, · · · , vk), then any suitablemaximum independent set L in G has the following properties.

(i) If vi ∈ L, then vi−2 ∈ L or there is a neighbor of vi−1 in V (Ti−1) ∩ L.(ii) If vj 6∈ L, then vj−1 ∈ L or V (Tj) ∩ L is a maximum independent set in

Tj.

Proof The first property comes directly from requirement (c) in the definition.Let us show that the second property is true as well. If vj 6∈ L and V (Tj) ∩L is not a maximum independent set in Tj , then (V (Tj) ∩ L) ∪ {vj} is, byrequirement (b). In this case, if vj−1 is not in L, then vj+1 must not in Leither for otherwise (L\{vj+1}) ∪ {vj} would be a maximum independent setin G, contradicting to requirement (c). Since both vj−1 and vj+1 are not inL, L ∪ {vj} becomes a larger independent set in G, which also leads to acontradiction. Therefore, we conclude that if vj 6∈ L and V (Tj) ∩ L is not amaximum independent set in Tj , then vj−1 must be in L. ut

Lemma 6 Let G be an odd unicycle graph with at least one Type I subtree oran even unicycle graph whose cycle is CG = (v1, v2, · · · , vk). If L is a suitablemaximum independent set in G, then there exists a set E′ of edges of G suchthat the following conditions are satisfied.

(i) Each edge in E′ connecting a vertex in L and a vertex not in L.(ii) Each vertex in L is on exactly one edge from E′ and each vertex in V (G)\L

is on some edge from E′.(iii) vivi+1 ∈ E′ for any vi ∈ L.(iv) If vj 6∈ L, then vj−1vj ∈ E′ where vj−1 ∈ L, or vjw ∈ E′ where w ∈

L ∩ V (Tj).


Proof For the given independent set L, we construct the required set E′ in thefollowing way. First, if vi ∈ L, then Ti is of Type II and therefore, by Lemma4, L ∩ V (Ti) is a maximum independent set in Ti + vi+1. Hence, any fan Fiof Ti + vi+1 associated with L ∩ V (Ti) must contain the edge vivi+1. Second,for vj 6∈ L, if L ∩ V (Tj) is a maximum independent set in Tj , then we let Fjbe any fan of Tj associated with L∩V (Tj). Fj certainly contains an edge vjwwith w ∈ L ∩ V (Tj). If L ∩ V (Tj) is not a maximum independent set in Tj ,then (L∩ V (Tj))∪ {vj} is and Tj is of Type II. In this case, Lemma 5 assertsthat vj−1 must be in L and Lemma 4 guarantees that (L ∩ V (Tj)) ∪ {vj} isalso a maximum independent set in Tj + vj−1. Thus any fan Fj of Tj + vj−1associated with (L ∩ V (Tj)) ∪ {vj} contains the edge vj−1vj which is also an

edge in Fj−1. Now, we let E′ =⋃ki=1 Fi. Since each Fj is a fan of a tree; each

vi ∈ L is on exactly one edge from E′ and each vj 6∈ L is on some edge fromE′ joining vj to some vertex in L, E′ has the required properties. ut

Note the set E′ is in fact a fan E(L) associated with L when G is an evenunicycle graph. Therefore we also call it a fan associated with the suitablemaximum independent set L when G is an odd unicycle graph with at leastone Type I subtree and denote it as E(L) in what follows. Now, a bound onAR(G) for a special case follows immediately.

Proposition 3 Let G be a unicycle graph of order n with the unique cycleCG. If G has a maximum independent set L such that V (CG) ∩ L = ∅, then

2− α(G) + 1

n≤ AR(G) ≤ 2− α(G)

n.

Proof Let CG = (v1, v2, · · · , vk). In this case, each subtree Tj must be of TypeI and L∩ V (Tj) is a maximum independent set in Tj . Hence a fan E(L) whichhas the properties in Lemma 6 contains an edge vjw with w ∈ L ∩ V (Tj) foreach j = 1, 2, · · · , k. Now, each V (H), where H is a component in G−E(L), isa core of G because the designated outside neighbor u of any vertex u ∈ V (H)can be chosen as the other endvertex of an edge uu in E(L) and all u’s areindependent. Since there are α(G)+1 components inG−E(L), c

∗(G) ≤ α(G)+1and the result follows by Theorem 21. ut

The next result has been shown to be true for bipartite graphs in Section4, we now show that it holds as well for odd unicycle graphs which have atleast one Type I subtree.

Lemma 7 Suppose that G is an odd unicycle graph with at least one TypeI subtree or an even unicycle graph whose cycle is CG = (v1, v2, · · · , vk) andE(L) is a fan obtained from Lemma 6 associated with a suitable maximumindependent set L in G. Let Π be an optimal star decomposition of G fromCorollary 7 and H be a component in G− E(L). If, for each u ∈ V (H), thereexists a neighbor u of u with uu ∈ E(L) such that u 6∈ V (H), then

(i) d−GΠ [V (H)](v) = dG(v) − 1 for each sink vertex v ∈ V (H) of GΠ and

d−GΠ [V (H)](u) = d−GΠ(u) for each nonsink vertex u ∈ V (H) of GΠ , and


(ii) m−V (H) = |{sink vertices of GΠ in V (H)}|.

Proof Since each v ∈ L∩V (H) is on exactly one edge vv in E(L) and v 6∈ V (H),

we have d−GΠ [V (H)](v) = dG(v) − 1. According to the orientation induced by

Π, each edge of GΠ going into a nonsink vertex u ∈ V (H) of GΠ does notbelong to E(L) because it connects two nonsink vertices of GΠ . So we have

d−GΠ [V (H)](u) = d−GΠ(u) for each nonsink vertex u ∈ V (H) of GΠ and therfore

m−V (H) = |{sink vertices of GΠ in V (H)}|. utNext, we propose some realizable unicycle graphs.

Theorem 23 Let G be an odd unicycle graph with at least one Type I subtreeor an even unicycle graph whose cycle is CG = (v1, v2, · · · , vk) and L be asuitable maximum independent set in G. Then G is realizable if one of thefollowing statements is true.

(i) |L ∩ V (CG)| ≥ 3;(ii) L ∩ V (CG) = {vi, vj} and d(vi, vj) ≥ 3;

(iii) L∩V (CG) = {vi, vi+2} and L∩V (Ti+3) is a maximum independent set inTi+3.

Proof Let E(L) be a fan associated with L satisfying the properties in Lemma6. Since there are α(G) components in G−E(L) in all three cases, by Theorem21, it suffices to show that {V (H)|H is a component in G − E(L)} is a corecluster of G. Note that, for vi ∈ L, vivi+1 is the unique incident edge ofvi in E(L). In cases (i) and (ii), for each component H in G − E(L), if we

choose eu to be an edge uu in E(L) and let E+H = {eu|u ∈ V (H)}, then the

extended graph H + E+H induces no cycles in G. G is realizable by Theorem

16, Lemma 7 and Theorem 9. Next, let us consider case (iii). In this case, thevi’s, i = 1, 2, · · · , i, i + 3, i + 4, · · · , k, are in the same component, say H. Itsuffices to show that there exists an extended graph of H whose vertex setinduces no cycles in G. Since L ∩ V (Ti+3) is a maximum independent set inTi+3, Ti+3 is nontrivial and we know from Lemma 6 that vi+3 is on someedge vi+3w ∈ E(L) where w ∈ V (Ti+3). Now, we let evi+3

be the edge vi+3w.For u ∈ V (H)\{vi+3}, we let eu be any incident edge of u in E(L). Sincevi+2 6∈ V (H +E+), where E+ = {ev|v ∈ V (H)}, the extended graph H +E+

induces no cycles in G and therefore G is realizable by Theorem 16, Lemma 7and Corollary 10. ut

The operation of replacing the edge e of G with a path of length k + 1 iscalled a k-subdivision of the edge e. This operation will be used in the proofof the next result.

Theorem 24 If G is a unicycle graph of order n with girth(G) ≥ 5, then

2− α(G) + 1

n≤ AR(G) ≤ 2− α(G)

n.

Proof Let CG = (v1, v2, · · · , vk) be the unique cycle of G. We consider the casewhere G is an odd unicycle graph with at least one Type I subtree or an even


unicycle graph first. We choose a fan E(L) obtained from Lemma 6 associatedwith a suitable maximum independent set L. By Proposition 3 and Theorem23, it suffices to only consider the following two cases.

Case (i): |L ∩ V (CG)| = 1. We may assume that L ∩ V (CG) = {vk}.Let {H1, H2, · · · , Hα(G)} be the components in G − E(L) and v1 ∈ H1, then

V (CG) ⊆ V (H1). Denote as V ′ the set V (H1)\{u ∈ V (Tj)|j = dk2 e+ 1, · · · , k}and as V ′′ the set V (H1)\V ′. Let us choose the designated outside neighbor uof any vertex u as the other endvetex of uu ∈ E(L). Observe that although vkand v1 are designated outside neighbors of each other, vd k2 e

and vd k2 e+1 have

designated outside neighbors in Td k2 eand Td k2 e+1 respectively by Lemma 6.

The set of all designated outside neighbors of the vertices in V ′ is independentin G, and so is the one for V ′′. Therefore, {V ′, V ′′, V (H2), · · · , V (Hα(G))} isa core cluster of G and c∗(G) ≤ α(G) + 1. The result follows by Thoerem 21.

Case (ii): |L ∩ V (CG)| = 2. By Theorem 23, we may assume that L ∩V (CG) = {vk, v2}. Let {H1, H2, · · · , Hα(G)} be the components inG−E(L) andv3 ∈ H1, then {vi|i = 3, · · · , k} ⊆ V (H1). Denote as V ′ the set V (H1)\{u ∈V (Tj)|j = dk2 e + 1, · · · , k} and as V ′′ the set V (H1)\V ′. By choosing thedesignated outside neighbor u of any vertex u as the other endvertex of uu ∈E(L), we have that {V ′, V ′′, V (H2), · · · , V (Hα(G))} is a core cluster of G andc∗(G) ≤ α(G) + 1.

Next, we consider odd unicycle graphs without Type I subtrees. Let us5-subdivide the edge vkv1 by replacing the edge with the path vk −w1−w2−w3 − w4 − w5 − v1 and denote the graph after subdivision as G′. Then G′ isa realizable even unicycle graph by Corollary 20 and there exists an optimalcore cluster C ′ of size α(G′). Let L′ be a suitable maximum independent setin G′. Since G′ does not contain Type I subtrees, we may assume that L′ ∩V (CG′) = {v1, v3, · · · , vk, w2, w4}, where CG′ = (v1, v2, · · · , vk, w1, · · · , w5).Since L′\{vk, w2, w4} is independent in G, α(G′) − 3 = |L′\{w2, w4, vk}| ≤α(G). From the way we construct an optimal core cluster for realizabe bipartitegraphs, we may ssume that C ′ = {V ′1 , V ′2 , · · · , V ′α(G′)} where V ′1 = {w1, w2},V ′2 = {w3, w4} and w5 ∈ V ′3 . By letting v1 and vk be the designated outsideneighbors of each other, C = {V ′3\{w5}, V ′4 , · · · , V ′α(G′)} is a core cluster of G

because v1 and vk lie in different cores. Hence c∗(G) ≤ α(G′) − 2. Therefore,we have c∗(G) + 2 ≤ α(G′) ≤ α(G) + 3, which implies that c∗(G) ≤ α(G) + 1.The result follows. ut

Next, we investigate this problem for unicycle graphs with smaller girth.

Theorem 25 Let G be a unicycle graph of order n with girth(G) = 4 and

θ(G) =∑4i=1 α(Ti), then AR(G) = 2 − θ(G)

n if G has at least two Tpye II

subtrees, and 2− θ(G)+1n ≤ AR(G) ≤ 2− θ(G)

n otherwise.

Proof Let CG = (v1, v2, v3, v4) be the unique cycle of G. First, we partition{1, 2, 3, 4} into two parts {x1, x∗1} and {x2, x∗2} such that vxi is a neighbor ofvx∗i and if G has at least two Type I subtrees, then each {Txi , Tx∗i }, i = 1, 2,must contain at least one of them. Next, among the part {xi, x∗i }, if there isany nontrivial Type II subtree in {Txi , Tx∗i }, then we let Txi be a nontrivial


Type II subtree and Tx∗i be the other; if there is no nontrivial Type II subtreein {Txi , Tx∗i }, then we let Txi be a Type I subtree. Only when there are nonontrivial Type II and Type I subtrees in {Txi , Tx∗i }, can Txi be a trivial TypeII subtree in {Txi , Tx∗i }. Therefore, there are several possible situations for thetypes of the subtrees (Txi , Tx∗i ):

(S1) (nontrivial Type II, trivial Type II);(S2) (nontrivial Type II, nontrivial Type II or Type I);(S3) (Type I, trivial Type II);(S4) (trivial Type II, trivial Type II);(S5) (Type I, Type I).

Now, we construct α(Txi) + α(Tx∗i ) cores to cover the vertices in V (Txi) ∪V (Tx∗i ) in situations (S1) to (S4), while in situations (S5) we construct α(Txi)+α(Tx∗i ) + 1 cores to cover the vertices in V (Txi) ∪ V (Tx∗i ) in the followingway. If Ti is nontrivial, we let Πi be an optimal star decomposition of Tiwith vertex-number sum nΠi and Ci = {Vi,1, Vi,2, · · · , Vi,α(Ti)+1} be an op-timal core cluster of Ti with vi ∈ Vi,1, then nΠi = 2|V (Ti)| − (α(Ti) + 1).In situation (S1), we consider an optimal core cluster C ′xi of Txi + vx∗i con-structed from Lemma 4, then |C ′xi | = α(Txi) + 1 = α(Txi) + α(Tx∗i ) becauseTx∗i is trivial. The cores in C ′xi cover the vertices in V (Txi) ∪ V (Tx∗i ). In sit-uation (S2), we may assume that the optimal core cluster of Txi + vx∗i isC ′xi = {V ′xi,1, V

′xi,2, · · · , V

′xi,α(Txi )+1} in which V ′xi,1 = {vx∗i } and vxi ∈ V ′xi,2.

Now, consider the set Vx∗i ,1 ∪ V′xi,2 where Vx∗i ,1 is the core containing vx∗i in

the optimal core cluster Cx∗i of Tx∗i . If we keep the original designated outsideneighbor for each vertex in Vx∗i ,1∪V

′xi,2 except vxi and let the one for vxi be the

neighbor of it in CG other than vx∗i , then Vx∗i ,1∪V′xi,2 is a core of G. Therefore,

Vx∗i ,1∪V′xi,2, V

′xi,3, · · · , V

′xi,α(Txi )+1, Vx∗i ,2, · · · , Vx∗i ,α(Tx∗i )+1 are α(Txi)+α(Tx∗i )

cores which cover the vertices in V (Txi) ∪ V (Tx∗i ).In situation (S3), Txi + vx∗i has an optimal core cluster C ′xi = {V ′xi,1, V

′xi,2,

· · · , V ′xi,α(Txi )+2} constructed from Lemma 3 which satisfies that V ′xi,1 =

{vx∗i }, vxi ∈ V′xi,2 and the designated outside neighbor of vxi is not vx∗i . By

assigning the designated outside neighbor of vx∗i as the neighbor of it in CGother than vxi , the set V ′xi,1 ∪ V

′xi,2 is a core of G. Since Tx∗i is trivial, the

sets V ′xi,1 ∪ V′xi,2, V

′xi,3, · · · , V

′xi,α(Txi )+2 are α(Txi) + α(Tx∗i ) cores which cover

the vertices in V (Txi) ∪ V (Tx∗i ). In situation (S4), since Txi and Tx∗i are bothtrivial, {vxi} and {vx∗i } are cores of G. We have α(Txi)+α(Tx∗i ) cores coveringthe vertices in V (Txi) ∪ V (Tx∗i ).

In situation (S5), let C ′xi = {V ′xi,1, V′xi,2, · · · , V

′xi,α(Txi )+2} be an optimal

core cluster of Txi + vx∗i constructed from Lemma 3 with V ′xi,1 = {vx∗i }, vxi ∈V ′xi,2 and the designated outside neighbor of vxi is not vx∗i . By assigning thedesignated outside neighbor of v∗xi as the one it has for the core Vx∗i ,1 in theoptimal core cluster Cx∗i of Tx∗i and keeping the original ones for the othervertices, Vx∗i ,1 ∪ V

′xi,2 is a core of G. We therefore have α(Txi) + α(Tx∗i ) + 1

cores Vx∗i ,1 ∪ V′xi,2, V

′xi,3, · · · , V

′xi,α(Txi )+2, Vx∗i ,2, · · · , Vx∗i ,α(Tx∗i )+1 which cover

the vertices in V (Txi) ∪ V (Tx∗i ).


In the case that G has at least two Type II subtrees, only situation (S5) isimpossible to occur because of the way we partition {1, 2, 3, 4} into {x1, x∗1}and {x2, x∗2} in the first place. Since we have constructed α(Txi)+α(Tx∗i ) corescovering the vertices in V (Txi) ∪ V (Tx∗i ) in situations (S1) to (S4), we have a

core cluster C of G which is of size∑2i=1(α(Txi) + α(Tx∗i )) =

∑4i=1 α(Ti) =

θ(G).

If G contains at least three Type I subtrees, then situation (S5) must occur.If all four are of Type I, then there exists a sutiable maximum independentset L in G such that V (CG)∩L = ∅ and α(G) = θ(G). The bound on AR(G)has been shown in Proposition 3. Now, we consider the case where G containsexactly three Type I subtrees. We may assume that (Tx1

, Tx∗1 ) is in situation(S5) and (Tx2

, Tx∗2 ) is not. In previous discussion, we have constructed α(Tx1)+

α(Tx∗1 ) + 1 cores covering V (Tx1)∪V (Tx∗1 ) and α(Tx2

) +α(Tx∗2 ) cores coveringV (Tx2) ∪ V (Tx∗2 ). Therefore, we have a core culster C of G which is of size∑4i=1 α(Ti) + 1 = θ(G) + 1.

On the other hand, we define a complete multipartite covering Π of G asΠ = {CG} ∪

(⋃i∈J Πi

)where J = {i|Ti is nontrivial}. Note that 2|V (Ti)| −

(α(Ti) + 1) = 0 holds for any trivial subtree Ti. The vertex-number sum of Πcan be evaluated as follows.

nΠ = 4 +∑i∈J

nΠi

= 4 +∑i∈J

(2|V (Ti)| − (α(Ti) + 1)) +∑

i∈{1,2,3,4}\J

(2|V (Ti)| − (α(Ti) + 1))

= 2

4∑i=1

|V (Ti)| −4∑i=1

α(Ti)

= 2|V (G)| − θ(G).

If G has at least two Type II subtrees, then nΠ = 2|V (G)|− |C | and therefore

G is realizable and AR(G) = 2− θ(G)n . If G has exactly three Type I subtrees,

then nΠ = 2|V (G)| − |C |+ 1 and we have the bound on AR(G). utNext, we consider unicycle graphs with a 3-cycle.

Theorem 26 Let G be a unicycle graph of order n with girth(G) = 3 and

θ(G) =∑3i=1 α(Ti), then AR(G) = 2 − θ(G)

n if every Ti is of Type II, and

2− θ(G)+1n ≤ AR(G) ≤ 2− θ(G)

n otherwise.

Proof Let CG = (v1, v2, v3) be the unique cycle of G. If Ti is nontrivial, welet Πi be an optimal star decomposition of Ti with vertex-number sum nΠi .We define a complete multipartite covering Π of G as Π = {CG}∪

(⋃i∈J Πi

),


where J = {i|Ti is nontrivial}. It has the vertex-number sum

nΠ = 3 +∑i∈J

nΠi

= 3 +∑i∈J

(2|V (Ti)| − (α(Ti) + 1)) +∑

i∈{1,2,3}\J

(2|V (Ti)| − (α(Ti) + 1))

= 2|V (G)| − θ(G).

Let C ′i = {V ′i,1, V ′i,2, · · · , V ′i,|C ′i |} be the optimal core cluster of Ti + vi+1

constructed from Lemma 3 or Lemma 4 in which V ′i,1 = {vi+1} and vi ∈ V ′i,2. IfTi is of Type I, then |C ′i | = α(Ti)+2. If Ti is of Type II, then |C ′i | = α(Ti)+1.Now, let us consider the following possible cases. (i) If all three subtrees Ti’sare of Type II, then C =

⋃i=1,2,3

(C ′i \V ′i,1

)is a core cluster of G of size

|C | =∑3i=1 α(Ti) = θ(G) and G is realizable. (ii) If T1 and T2 are of Type

II and T3 is of Type I, then C =⋃i=1,2,3

(C ′i \V ′i,1

)is a core cluster of G of

size |C | =∑3i=1 α(Ti) + 1 = θ(G) + 1 and the bound on AR(G) follows. (iii)

If T1 is of Type II and T2 and T3 are not, then V ′2,2 ∪ V ′3,2 is a core of G and(C ′1\{V ′1,1}

)∪(⋃

i=2,3

(C ′i \{V ′i,1, V ′i,2}

))∪ {V ′2,2 ∪ V ′3,2} is a core cluster of G

of size |C | =∑3i=1 α(Ti) + 1 = θ(G) + 1 and we have the bound on AR(G)

as well. (iv) If all three subtrees are of Type I, then there exists a maximumindependent set L in G such that L ∩ CG = ∅ and θ(G) = α(G). The resultfollows by Proposition 3. Our proof is completed. ut

We summarize our results on the optimal average information ratio ofunicycle graphs in the next theorem.

Theorem 27 Let G be a unicycle graph of order n with a cycle CG = (v1, v2,· · · , vk), then

2− θ(G) + 1

n≤ AR(G) ≤ 2− θ(G)

n,

where θ(G) = α(G) if girth(G) ≥ 5 and θ(G) =∑ki=1 α(Ti) otherwise.

Furthermore, AR(G) = 2− θ(G)n if one of the following conditions is satisfied.

(i) G is an odd unicycle graph with at least one Type I subtree or an evenunicycle graph and G has a suitable maximum independent set L such thatone of the following statements is true.

(a) |L ∩ V (CG)| ≥ 3;(b) L ∩ V (CG) = {vi, vj} and d(vi, vj) ≥ 3;(c) L∩V (CG) = {vi, vi+2} and L∩V (Ti+3) is a maximum independent set

in Ti+3.(ii) k = 4 and G has at least two Tpye II subtrees;

(iii) k = 3 and every Ti is of Type II.


6 Conclusion

In this paper, we show the existence of optimal star decompositions and es-tablish bounds on the optimal average information ratio for general graphs.We also propose conditions under which the value of the optimal average in-formation ratio can be determined and subsequently some infinite classes ofrealizable bipartite graphs are introduced. This extends the main result in[19]. Furthermore, we determine the exact vlaues of the optimal average infor-mation ratio of some infinite classes of unicycle graphs and also give a goodbound on the ratio for all unicycle graphs. The difference between our upperbound and lower bound is very small especially when n is large. It is worthnoting that the exact value of the optimal average information ratio also serveas a lower bound on the unknown optimal information ratio for those graphs.Deriving meaningful lower bounds on the optimal (average) information ratiois in general much more difficult than deriving upper bounds for most graphs.

Based on the discussion in this paper, one can see that constructing agood core cluster of G is essential in the problem of invesgating the valueof the optimal average information ratio of the graph G. Besides, for graphsof small girth, star coverings may not result in the least vertex-number sum.Some other complete multipartite subgraphs of G must be used in a completemultipartite covering to achieve the best results. Both of these questions arechallenging and also worth trying.

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the optimal average information ratio of secret-sharing ......over all possible secret-sharing...

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