contractions of planar graphs - rutgers center for operations research
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CONTRACTIONS OF PLANAR GRAPHS
ESA 2010
Marcin KamińskiBrussels
Daniël PaulusmaDurham
Dimitrios ThilikosAthens
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CONTRACTIONS OF PLANAR GRAPHS2
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CONTAINMENT RELATIONS
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induced subgraph ✓ ✗ ✗
subgraph ✓ ✓ ✗
minor ✓ ✓ ✓
contraction ✗ ✗ ✓
induced minor ✓ ✗ ✓
CONTAINMENT RELATIONS
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CONTRACTIONS OF PLANAR GRAPHS4
PREVIOUS WORK
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CONTRACTIONS ALGORITHMICALLY
Theorem (Matoušek and Thomas, 1992)
The problem of deciding, given two input graphs G and H, whether G is contractible to H is NP-complete even if H and G are trees:
of bounded diameter; or, all whose vertices but one have degree at most 5.
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CONTRACTIONS ALGORITHMICALLY
H-contractibility
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CONTRACTIONS ALGORITHMICALLY
H-contractibility
Theorem (Brouwer and Veldman, 1987)
Let H be a triangle-free graph. If H is a star, then H-contractibility is in P; otherwise is NP-complete.
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CONTRACTIONS ALGORITHMICALLY
H-contractibility
Theorem (Brouwer and Veldman, 1987)
Let H be a triangle-free graph. If H is a star, then H-contractibility is in P; otherwise is NP-complete.
Remark
P4-contractibility, C4-contractibility are NP-complete problems.
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CONTRACTIONS ALGORITHMICALLY
Theorem (Levin, Paulusma, and Woeginger, 2002)
Let H be a connected graph on at most 5 vertices.
If H has a dominating vertex, then H-contractibility is in P. If H does not have a dominating vertex, then H-contractibility is NP-complete.
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CONTRACTIONS ALGORITHMICALLY
Theorem (Levin, Paulusma, and Woeginger, 2002)
Let H be a connected graph on at most 5 vertices.
If H has a dominating vertex, then H-contractibility is in P. If H does not have a dominating vertex, then H-contractibility is NP-complete.
Observation (van ’t Hof, Kamiński, Paulusma, Szeider, and Thilikos, 2009)
There exists a graph H on 69 vertices with a dominating vertex for which H-contractibility is NP-complete.
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CONTRACTIONS ALGORITHMICALLY
Theorem (Matoušek, Nešetril, and Thomas, 1988)
There exists a non-recursive class of graphs closed under taking of contractions (and induced subgraphs).
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OUR RESULTS
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MAIN THEOREM
Theorem
For every graph H, there exists a polynomial-time algorithm, deciding whether the planar input graph is contractible to H.
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COMBINATORIALLY EQUIVALENT
Two plane graphs G and H are combinatorially equivalent if there exists a homeomorphism of the unit sphere (in which they are embedded) which transforms one into the other.
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THIN GRAPHS
Homotopic edges = edges bounding a 2-face
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THIN GRAPHS
Homotopic edges = edges bounding a 2-face
Thin graph = a plane multigraph without homotopic pairs of edges
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CONTRACTIONS OF PLANAR GRAPHS17
THIN GRAPHS
Homotopic edges = edges bounding a 2-face
Thin graph = a plane multigraph without homotopic pairs of edges
Lemma (Alber, Fellows, and Niedermeier, 2004)
If G is a thin graph, then |E(G)| ≤ 3|V(G)| - 6.
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EMBEDDED CONTAINMENT RELATIONS
Contraction (≤c) and embedded contraction (≤ec).
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EMBEDDED CONTAINMENT RELATIONS
Contraction (≤c) and embedded contraction (≤ec).
Dissolution and embedded dissolution.
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EMBEDDED CONTAINMENT RELATIONS
Contraction (≤c) and embedded contraction (≤ec)
Dissolution and embedded dissolution.
Topological minor (≤tm) and embedded topological minor (≤etm).
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EMBEDDED CONTAINMENT RELATIONS
Theorem
Let H and G be two thin graphs and H*, G* their respective duals.
H ≤ec G ⟺ H* ≤etm G*
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PATTERNS
A simple planar graph H is a pattern of a planar multigraph H’, if
V(H) = V(H’), and two vertices are adjacent in H iff they are adjacent in H’.
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PATTERNS
A simple planar graph H is a pattern of a planar multigraph H’, if
V(H) = V(H’), and two vertices are adjacent in H iff they are adjacent in H’.
C(H) = a maximal set of all combinatorially different thin plane multigraphs whose pattern is H
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PATTERNS
A simple planar graph H is a pattern of a planar multigraph H’, if
V(H) = V(H’), and two vertices are adjacent in H iff they are adjacent in H’.
C(H) = a maximal set of all combinatorially different thin plane multigraphs whose pattern is H
Lemma
For every planar graph H, the set C(H) is finite.
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CONTRACTIONS AND EMBEDDED TOPOLOGICAL MINORS
Theorem
Let H and G be simple planar graphs and G be a plane graph isomorphic to G.
H ≤c G ⟺ ∃ H∊C(H) such that H ≤ec G
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CONTRACTIONS AND EMBEDDED TOPOLOGICAL MINORS
Theorem
Let H and G be simple planar graphs and G be a plane graph isomorphic to G.
H ≤c G ⟺ ∃ H∊C(H) such that H ≤ec G
Corollary
Let H and G be simple planar graphs and G be a plane graph isomorphic to G.
H ≤c G ⟺ ∃ H∊C(H) such that H* ≤etm G*
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TESTING FOR EMBEDDED TOPOLOGICAL MINORS
Reduction to testing for a collection of disjoint paths.
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TESTING FOR EMBEDDED TOPOLOGICAL MINORS
Reduction to testing for a collection of disjoint paths.
Theorem (Robertson and Seymour, 1995)
There exists an algorithm that given a graph G and k pairs (s1, t1), ..., (sk, tk) of vertices of G decides whether there are k vertex-disjoint paths P1, ..., Pk in G such that Pi joins si, ti, for all i=1, ..., k, and if so, finds them. The algorithm runs in O(|V(G)|3).
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TESTING FOR EMBEDDED TOPOLOGICAL MINORS
Topological minors.
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TESTING FOR EMBEDDED TOPOLOGICAL MINORS
Topological minors.
Embedded topological minors. Cyclic order of paths/neighbors.
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TESTING FOR EMBEDDED TOPOLOGICAL MINORS
Topological minors.
Embedded topological minors. Cyclic order of paths/neighbors.
|V(G)|O(|V(H)|)
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TESTING FOR EMBEDDED TOPOLOGICAL MINORS
Topological minors.
Embedded topological minors. Cyclic order of paths/neighbors.
|V(G)|O(|V(H)|)
Open problem
What is the parameterized complexity of deciding whether H is a topological minor of a (planar) input graph G, when parameterized by |V(H)|? FPT or W[1]-hard?
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MAIN THEOREM
Theorem
For every graph H, there exists a polynomial-time algorithm, deciding whether the planar input graph is contractible to H.
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MAIN THEOREM
Theorem
For every graph H, there exists a polynomial-time algorithm, deciding whether the planar input graph is contractible to H.
Generalization to bounded genus graphs.
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MAIN THEOREM
Theorem
For every graph H, there exists a polynomial-time algorithm, deciding whether the planar input graph is contractible to H.
Generalization to bounded genus graphs.
Theorem
For every integer g≥0 and a graph H, there exists a polynomial-time algorithm, deciding whether the input graph, which is embeddable on a surface of Euler genus g, is contractible to H.
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CYCLICITY
cyclicity of G = the largest integer k for which G is contractible to Ck
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CYCLICITY
cyclicity of G = the largest integer k for which G is contractible to Ck
Theorem (Hammack, 1999)
There exists a polynomial-time algorithm to determine the cyclicity of a planar graph.
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CYCLICITY
cyclicity of G = the largest integer k for which G is contractible to Ck
Theorem (Hammack, 1999)
There exists a polynomial-time algorithm to determine the cyclicity of a planar graph.
Generalization to bounded genus graphs.
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CONTRACTIONS OF PLANAR GRAPHS
THANK YOU!
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