Minimal Dominating Sets in Graphs:Enumeration, Combinatorial Bounds and

Graph Classes

J.-F. Couturier1 P. Heggernes2

D. Kratsch1 P. van t’ Hof2

1LITAUniversite de Lorraine

F-57045 MetzFrance

2University of BergenN-5020 Bergen

Norway

Journees Combinatoires Rhones-Alpes AuvergneClermont-Ferrand, 1-2 fevrier, 2012

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Contents

Exact Exponential Algorithms

Input-Sensitive Enumeration

Known Algorithms and Combinatorial Bounds

Dominating Set

Graph ClassesChordal GraphsCographsChain GraphsCobipartite Graphs

Open Questions

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I. Solving computationally hard problemsexactly

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How to attack NP-hard problems

Various techniques have been developed to attack NP-hardproblems :

I approximation algorithms

I heuristics

I parameterized algorithms

I randomized algorithms

I restricted inputs

I exact exponential algorithms

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Exact exponential-time algorithms

I dates back to the early nineteen sixties

I Davis, Putnam (1960) and Bellmann ; Held, Karp (1962)

I tries to cope with NP-completeness in a strong sense

I solves problem exactly

I worst-case analysis of running time (and space)

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Why Study Exponential Algorithms ?

I leads to a better understanding of NP-hard problems

I initiates interesting new combinatorial and algorithmicchallenges

I Alan Perlis (first Turing Award winner) : “for everypolynomial-time algorithm you have, there is an exponentialalgorithm that I would rather run”.

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II. Why study input-sensitive enumerationalgorithms ?

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Types of problems

General

I decision

I optimization

I counting

I enumeration

Example

I decision problem : Has the graph G a dominating set ?

I optimization problem : Find the minimum size of adominating set of G .

I counting problem : Count the number of dominating sets of Gof size k .

I enumeration problem : List all minimal dominating sets of G .

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An input-sensitive algorithm solving an enumerationproblem ...

I can be used to solve appropriate decision, optimization andcounting versions of the problem

I can be used to solve other NP-hard problems

I provides an upper bound for the maximum number ofenumerated objects

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What is the maximum number of vertex sets satisfying agiven property in an n-vertex graph ?

Upper bounds ...

I are crucial for the analysis of the worst-case running time ofexact algorithms

I allow to use enumeration algorithms as subroutines

I may lead to matching lower and upper bounds (up to apolynomial factor)

I are of interest in their own in Combinatorics

Examples

I What is the maximum number of maximal independent sets inan n-vertex graph ?

I What is the maximum number of minimal dominating sets inan n-vertex graph ?

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III. Highlights

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Maximal Independent Sets

Moon,Moser 1965

I The maximum number of maximal independent sets in anygraph on n vertices is 3n/3.

I Original proof by induction

I Easy to transform into a branching algorithm enumerating allmaximal independent sets of a graph in time O∗(3n/3)

I Polynomial-delay algorithm (Tsukiyama, Ide, Ariyoshi,Shirakawa, 1977)

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Application in Coloring Algorithms

I based on dynamic programming over set of all maximalindependent sets

I O(2.4422n) time optimal coloring algorithm (Lawler 1976)

I improved upper bound on the maximum number of maximalindependent sets of small size

I O(2.4150n) time algorithm to compute an optimal coloring ofa graph (Eppstein 2003)

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Minimal Feedback Vertex Sets

Fomin,Gaspers,Pyatkin,Razgon, 2008

I The maximum number of minimal feedback vertex sets in anygraph on n vertices is at most 1.8638n.

I Proof based on a branching algorithm to enumerate allminimal feedback vertex sets of running time O(1.8638n).

I Lower Bound : There exists an infinite family of graphs allhaving 105n/10 ≈ 1.5926n minimal feedback vertex sets.

I Polynomial-delay algorithm (Schwikowski, Speckenmeyer,2002)

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Minimal Separators and Potential Maximal Cliques

Fomin,Villanger, ICALP 2008 and STACS 2010

I There is an algorithm to enumerate all potential maximalcliques of a graph in time O(1.734601n).

I The maximum number of potential maximal cliques in anygraph on n vertices is at most O(1.734601n).

I Every n-vertex graph has at most O(1.6181n) minimalseparators.

I Lower Bound : There exists an infinite family of graphs allhaving 3n/3 ≈ 1.4422n minimal separators and 3n/3 ≈ 1.4422n

maximal potential cliques.

Application in Treewidth Algorithms

I O(1.7549n) algorithm to exactly compute the treewidth of agraph. (Fomin,Villanger, ICALP 2008)

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IV. Dominating Set

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Fundamental notions

dominating set

A vertex set S ⊆ V is a dominating set of G if every vertex of Geither belongs to S or has a neighbour in S . Every vertex v of adominating set dominates the vertices in N[v ].

minimal dominating set (mds)

A dominating set S is a minimal dominating set (mds) if no propersubset of S is a dominating set.

private neighbour

I If S is a mds, then for every vertex v ∈ S , there is a vertexx ∈ N[v ] which is dominated only by v .

I Such a vertex x is a private neighbor of v , since x is notadjacent to any vertex in S \ {v}.

I A vertex in S is its own private neighbor if it is an isolate inG [S ].

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Number of minimal dominating sets

We denote by µ(G ) the number of mds of G .

LemmaLet G be a graph with connected components G1, G2, . . . , Gt .Then µ(G ) =

∏ti=1 µ(Gi ).

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Enumerating minimal dominating sets

Fomin,Grandoni,Pyatkin,Stepanov, 2008

I The maximum number of minimal dominating sets in anygraph on n vertices is at most 1.7159n.

I Proof based on a branching algorithm to enumerate allminimal dominating sets of running time O(1.7159n).

I Lower Bound : There exists an infinite family of graphs allhaving 15n/6 ≈ 1.5704n minimal dominating sets.

I No algorithm of output-polynomial running time known

I No polynomial delay algorithm known

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V. Graph Classes

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Our Goals

I We provide upper and lower bounds for the maximum numberof minimal dominating sets

I in chordal, cographs, chain, cobipartite graphs ...

I Typically we have either matching upper and lower bounds,using combinatorial arguments

I or asymptotic bounds proved by using branching algorithms.

I We obtain algorithms to enumerate all minimal dominatingsets for graphs in each of our graph classes.

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Summary of Upper and Lower Bounds

Graph Class Lower Bound Upper Bound

general [FGPS08] 1.5704n 1.7159n

chordal 1.4422n 1.6181n

cobipartite 1.3195n 1.5875n

split 1.4422n 1.4656n

proper interval 1.4422n 1.4656n

cograph∗ 1.5704n 1.5705n

trivially perfect∗ 1.4422n 1.4423n

threshold∗ ω(G ) ω(G )

chain∗ bn/2c+ m bn/2c+ m

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Some Lower Bounds

Cographs

I µ(3K2) = 15 ⇒ µ(t3K2) = 15t

I There exists an infinite family of cographs all having15n/6 ≈ 1.5704n minimal dominating sets.

Chordal Graphs

I µ(K3) = 3 ⇒ µ(tK3) = 3t

I There exists an infinite family of chordal graphs all having3n/3 ≈ 1.4422n minimal dominating sets.

I Disjoint union of triangles also proper interval, triviallyperfect, cograph, permutation graph, ...

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Output-Sensitive Enumeration

Kante, Limouzy, Mary, Nourine, FCT 2011

I Study of the output-sensitive complexity of enumerating allminimal dominating sets in graph classes

I Polynomial-delay algorithm for split graphs

I Further classes with polynomial delay algorithm

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Branching Algorithms

I Branching algorithms

recursively applied to instances of the problem usingBranching rules and Reduction rules.

I Branching rules : solving the problem by recursively solvingsmaller instances

I Reduction rules :

- simplify the instance- (typically) reduce the size of the instance

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Search Trees

I Search Tree :

used to illustrate and analyse an execution of a Branchingalgorithm

I nodes : assigns to each node a solved problem instance

I root : assigns the input to the root

I child : each instance (of a subproblem) reached by a branchingrule is assigned to a child of the node (of the original instance)

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A search tree

Select Discard

S

S S

S D

D

D

D

V1

V2 V4

V3 V5

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Analysis of the Running Time

To obtain an upper bound on the maximum number of nodes ofthe search tree (for an input of size n) :

1. Define a Measure for a problem instance.

2. Lower bound the progress made by the algorithm at eachbranching step.

3. Compute the collection of linear recurrences and branchingvectors by considering all branching rules.

4. Solve all those recurrences by the computation of thebranching number (to obtain an upper bound of the formO(αn

i )) for each one.

5. Take the worst case over all solutions : maximum branchingnumber α.

Then the maximum number of leaves in any search tree is O(αn).

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V.1. Chordal Graphs

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Enumeration via Branching

Main Features

I branching algorithm generates vertex subsets at the leaves ofthe corresponding search tree

I every minimal dominating set is assigned to a leaf of thesearch tree

I ALLOWED : multiple occurences of mds as well as vertex setsnot being mds

I (at termination simply check each leaf whether its vertex setis indeed a mds)

Thus an upper bound on the maximum number of leaves in thesearch tree for input graph G is also an upper bound for µ(G ), i.e.,the number of mds of G .

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Setting of our Algorithm and its Analysis

Invariants

I INSTANCE of a recursiv call : (G ′,D) where G ′ is an inducedsubgraph of the input graph G , and D is a subset ofV (G ) \ V (G ′).

I D chosen for mds, i.e., D subset of any vertex set of a leaf inthe subtree rooted at (G ′,D)

I D dominates all vertices of V (G ) \ V (G ′) ; thus any vertex ofG ′ when added to D must have a private neighbour in G ′.

Features

I Choice of vertex to branch based on structural properties

I Measure of an instance (G ′,D) is defined by assigning aweight of 0 or 1 to each vertex and by then taking the sum ofthe weights over V

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Structural Properties of Chordal Graphs

Definitions

I A graph G is chordal if every cycle of length at least 4 has achord.

I A vertex x of G is simplicial if its neighbourhood N(x) is aclique.

Properties

I Every chordal graph has a simplicial vertex.

I If x is a simplicial vertex of G then for all y ∈ N(x),N[x ] ⊆ N[y ].

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Branching 0

(G ′,D) : Choose simplicial vertex x of G ′ to branch on.

I Vertices of G ′ are dominated or not, and forbidden or not.

I Reduction rule : x isolated.

I There are 3 different branching rules.

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Branching 1

Case 1 : x is already dominated.

I x ∈ D. Since x is simplicial and needs a private neighbour inN(x), we can delete x and all its neighbours.

I x /∈ D. Since it is already dominated, it is safe to delete x .

Branching vector (2, 1)

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Branching 2

Case 2 : x is not already dominated and |N(x)| ≥ 2

Let y be a neigbour of x .

I y ∈ D. Since x is simplicial, all neighbours of x are dominatedby y . We delete x and y .

I y /∈ D. Since y ∈ N(x), any vertex we select later to dominatex will also dominate y . Thus we can delete y .

Branching vector (2, 1)35/54

Branching 3

Case 3 : x is not already dominated and |N(x)| = 1.

Let y be the neighbour of x .

I x ∈ D. Since y is the private neighbour of x , we can delete xand y .

I x /∈ D. The only way to dominate x is to take y into D.Hence y ∈ D and we can delete x and y .

Branching vector (2, 2)

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Analysis

Running Time

I Branching vectors : (2, 1), (2, 1), (2, 2)

I τ(2, 1) ≈ 1.6181, τ(2, 2) ≈ 1.4142

I Algorithm to enumerate all mds of a chordal graph runs intime O(1.6181n).

Bounds

I The maximum number of minimal dominating sets in achordal graph on n vertices is at most O(1.6181n).

I Lower bound : 1.4422n

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V.2. Cographs

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Structural Properties of Cographs

Definitions

I The disjoint union of G1 and G2 is the graphG1 ] G2 = (V1 ∪ V2,E1 ∪ E2).

I The join of G1 and G2 is the graphG1 on G2 = (V1 ∪ V2,E1 ∪ E2 ∪ {v1v2 | v1 ∈ V1, v2 ∈ V2}).

I A graph G is a cograph if it can be constructed from isolatedvertices by the operations disjoint union and join.

Properties

I Every cograph can be represented by its cotree.

I A graph is a cograph iff it has no P4 as induced subgraph.

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Proof by induction

TheoremEvery cograph has at most 15

n6 minimal dominating sets.

Proof by induction.

It is easy to enumerate all the cographs with n ≤ 6 vertices and toverify that each has at most 15

n6 minimal dominating sets.

Assume the theorem is true for all cographs with less than nvertices. Let G be a cograph with n vertices.

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Proof by induction

TheoremEvery cograph has at most 15

n6 minimal dominating sets.

Proof by induction.

It is easy to enumerate all the cographs with n ≤ 6 vertices and toverify that each has at most 15

n6 minimal dominating sets.

Assume the theorem is true for all cographs with less than nvertices. Let G be a cograph with n vertices.

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Decomposition

Decomposition

I Since every cograph G can be constructed from isolatedvertices by disjoint union and by join operation ...

I Cograph G can be decomposed into graphs G1 with n1vertices and G2 with n2 vertices such that :

1. either G is a disjoint union of G1 and G2 and there is no edgebetween G1 and G2.

2. or G is a join of G1 and G2 and all edges with one endpoint inG1 and one in G2 are present.

I Clearly n = n1 + n2.

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G = G1 ] G2

I µ(G ) = µ(G1) · µ(G2)

I By induction hypothesis µ(G ) ≤ 15n16 · 15

n26 = 15

n6

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G = G1 on G2

Since x1x2 ∈ E (G ) for all x1 ∈ V (G1) and for all x2 ∈ V (G2), thereare three types of minimal dominating sets of G :

I minimal dominating set D1 of G1,

I minimal dominating set D2 of G2,

I {x1, x2}, for all x1 ∈ V (G1) and for all x2 ∈ V (G2)

I Consequently µ(G ) = µ(G1) + µ(G2) + n1 · n2.

I By induction hypothesis and since n ≥ 7,µ(G ) ≤ 15

n16 + 15

n26 + n1 · n2 ≤ 15

n6 .

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Upper Bound Matches Lower Bound

The maximum number of minimal dominating sets in a cograph onn = 6t vertices is 15n/6.

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V.3. Chain Graphs

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Structural Properties of Chain Graphs

Definitions

I A bipartite graph G = (A,B,E ) is a chain graph if there is anordering σA = 〈a1, a2, . . . , ak〉 of the vertices of A such thatN(a1) ⊆ N(a2) ⊆ · · · ⊆ N(ak), as well as an orderingσB = 〈b1, b2, . . . , b`〉 of the vertices of B such thatN(b1) ⊇ N(b2) ⊇ · · · ⊇ N(b`).

I σA and σB together chain ordering of G

Properties

I In a disconnected chain graph at most one componentcontains edges.

I All isolated vertices belong to every minimal dominating set.(and every maximal independent set).

I W.l.o.g. chain graph G connected.

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A chain graph with n/2 + 1 maximal independent sets

a1 a2 a3 a4 a5

b1 b2 b3 b4 b5

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The exact value of the maximum number of mis

Maximum number of misA chain graph has at most bn/2c+ 1 maximal independent sets,and there are chain graphs that have bn/2c+ 1 maximalindependent sets.

Lower Bound

I For every even n ≥ 2, let Gn be the chain graph obtained fromtwo independent sets A = {a1, . . . , an/2} andB = {b1, . . . , bn/2} by making ai adjacent to every vertex in{b1, . . . , bi}, for i = 1, . . . , n/2.

I For every even n ≥ 2, the graph Gn contains exactly bn/2c+ 1maximal independent sets.

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Structure of minimal dominating sets

At most one edge in mds

For every minimal dominating set S of a chain graph G , the graphG [S ] contains at most one edge.

Exactly one mds containing edge ab

I Let ab be an edge of a chain graph G = (A,B,E ) with a ∈ Aand b ∈ B.

I If a or b has degree 1, then there is no minimal dominatingset in G containing both a and b.

I If both a and b have degree at least 2, then there is exactlyone minimal dominating set in G containing both a and b.

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The exact value of the maximum number of mis

Maximum number of minimal dominating sets

A chain graph on at least 2 vertices has at most bn/2c+ mminimal dominating sets, and there are chain graphs withbn/2c+ m minimal dominating sets.

Lower Bound

I Take graph Gn from lower bound for mis, n even.

I σA = 〈a1, . . . , ak〉 and σB = 〈b1, . . . , b`〉 chain ordering of Gn

I Let G ′n be the graph obtained from Gn by adding the edgeak−1b`.

I The graph G ′n contains exactly bn/2c+ m mds : one for eachof the bn/2c maximal independent sets, and one for each edgeof G ′n, apart from the edge a1b1.

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V.4. Cobipartite Graphs

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Definition

I A graph G = (V ,E ) is cobipartite if its vertex set can bepartitioned into two cliques X and Y .

Lower Bound

I For n = 5k , two disjoint cliques X and Y , where |X | = k and|Y | = 4k .

I Make every vertex in X adjacent to exactly four vertices in Y ,such that every vertex in Y is adjacent to exactly one vertexin X .

I The graph has 4n/5 ≈ 1.3195n mds being subsets of Y .

Upper Bound

A cobipartite graph on n vertices has at most O(1.5875n) minimaldominating sets.

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Upper Bound

I |X | = αn with 0.5 ≤ α ≤ 1, and |Y | = (1− α)n

I polynomial number of mds D with |D| = 1, or |D ∩ X | = 1and |D ∩ Y | = 1

I number of mds S ⊆ Y is 2|Y | ≤ 2n/2

I How many mds D ⊆ X ? Let |D| = βn, where β ≤ α and2 ≤ βn ≤ |X |.

1. every vertex of D ⊆ X has a private neighbour which must befrom Y ...

2. implies β ≤ 1− α3. number of subsets of X of size βn is

(αnβn

)4. for fixed α maximizes with β = α/2 :

(αn

α/2 n

)5. to optimally choose α : β = α/2 ≤ 1− α implies α ≤ 2/3.6. Hence number of mds D, with |D| ≥ 2 and D ⊆ X , is at most(2n/3

n/3

), which is less than or equal to 22n/3

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VII. Open Questions

I Conjecture : maximum number of mds in a general graph is15n/6.

I Conjecture : maximum number of mds in interval and splitgraphs is 3n/3.

I Bipartite graphs : known lower bound 6n/4 ≈ 1.5650n, disjointunion of C4.Find an upper bound better than 1.7159n.

I Find an output-polynomial algorithm to enumerate all mds ofchordal graphs.

I Find an output-polynomial algorithm to enumerate all mds ofgraphs in general.

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