CONTRACTIONS OF PLANAR GRAPHS - Rutgers Center for Operations Research

CONTRACTIONS OF PLANAR GRAPHS

ESA 2010

Marcin KamińskiBrussels

Daniël PaulusmaDurham

Dimitrios ThilikosAthens

Saturday, May 28, 2011

CONTRACTIONS OF PLANAR GRAPHS2

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CONTAINMENT RELATIONS



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induced subgraph ✓ ✗ ✗

subgraph ✓ ✓ ✗

minor ✓ ✓ ✓

contraction ✗ ✗ ✓

induced minor ✓ ✗ ✓

CONTAINMENT RELATIONS



PREVIOUS WORK



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.




H-contractibility




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.




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.




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.




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.




Theorem (Matoušek, Nešetril, and Thomas, 1988)

There exists a non-recursive class of graphs closed under taking of contractions (and induced subgraphs).



OUR RESULTS



MAIN THEOREM

Theorem

For every graph H, there exists a polynomial-time algorithm, deciding whether the planar input graph is contractible to H.



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.



THIN GRAPHS

Homotopic edges = edges bounding a 2-face



THIN GRAPHS


Thin graph = a plane multigraph without homotopic pairs of edges



THIN GRAPHS


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.



EMBEDDED CONTAINMENT RELATIONS

Contraction (≤c) and embedded contraction (≤ec).




Contraction (≤c) and embedded contraction (≤ec).

Dissolution and embedded dissolution.




Contraction (≤c) and embedded contraction (≤ec)

Dissolution and embedded dissolution.

Topological minor (≤tm) and embedded topological minor (≤etm).




Theorem

Let H and G be two thin graphs and H*, G* their respective duals.

H ≤ec G ⟺ H* ≤etm G*



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’.



PATTERNS



C(H) = a maximal set of all combinatorially different thin plane multigraphs whose pattern is H



PATTERNS



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.



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



CONTRACTIONS AND EMBEDDED TOPOLOGICAL MINORS

Theorem


H ≤c G ⟺ ∃ H∊C(H) such that H ≤ec G

Corollary


H ≤c G ⟺ ∃ H∊C(H) such that H* ≤etm G*



TESTING FOR EMBEDDED TOPOLOGICAL MINORS

Reduction to testing for a collection of disjoint paths.




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).




Topological minors.




Topological minors.

Embedded topological minors. Cyclic order of paths/neighbors.




Topological minors.


|V(G)|O(|V(H)|)




Topological minors.


|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?



MAIN THEOREM

Theorem




MAIN THEOREM

Theorem


Generalization to bounded genus graphs.



MAIN THEOREM

Theorem



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.



CYCLICITY

cyclicity of G = the largest integer k for which G is contractible to Ck



CYCLICITY


Theorem (Hammack, 1999)

There exists a polynomial-time algorithm to determine the cyclicity of a planar graph.


CONTRACTIONS OF PLANAR GRAPHS

THANK YOU!


CONTRACTIONS OF PLANAR GRAPHS - Rutgers Center for Operations Research

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