[Paper Review] Hereditary Graph Classes: When the Complexities of Colouring and Clique Cover Coincide

Alexandre Blanché, Konrad K. Dabrowski+2 more|arXiv (Cornell University)|Jul 22, 2016

Advanced Graph Theory Research32 references3 citations

TL;DR

This paper classifies the computational complexity of Colouring and Clique Cover for $(H, \overline{H})$-free graphs, showing that both problems have the same complexity due to complement-closure. It resolves most cases, leaving only $sP_1 + P_3$ for $s \geq 3$ and $sP_1 + P_4$ for $s \geq 2$ open, and establishes NP-completeness for 5-Colouring in $(P_6, \overline{P_1 + P_6})$-free graphs.

ABSTRACT

A graph is $(H_1,H_2)$-free for a pair of graphs $H_1,H_2$ if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. In 2001, Král', Kratochv\'ıl, Tuza, and Woeginger initiated a study into the complexity of Colouring for $(H_1,H_2)$-free graphs. Since then, others have tried to complete their study, but many cases remain open. We focus on those $(H_1,H_2)$-free graphs where $H_2$ is $\overline{H_1}$, the complement of $H_1$. As these classes are closed under complementation, the computational complexities of Colouring and Clique Cover coincide. By combining new and known results, we are able to classify the complexity of Colouring and Clique Cover for $(H,\overline{H})$-free graphs for all cases except when $H=sP_1+ P_3$ for $s\geq 3$ or $H=sP_1+P_4$ for $s\geq 2$. We also classify the complexity of Colouring on graph classes characterized by forbidding a finite number of self-complementary induced subgraphs, and we initiate a study of $k$-Colouring for $(P_r,\overline{P_r})$-free graphs.

Motivation & Objective

• To classify the complexity of Colouring and Clique Cover for $(H, \overline{H})$-free graphs, where the two problems coincide due to complement-closure.
• To resolve open cases in the complexity classification of Colouring for $(H_1, H_2)$-free graphs when $H_2 = \overline{H_1}$ and $H_1$ is not self-complementary.
• To extend the classification to graph classes defined by forbidding a finite set of self-complementary induced subgraphs.
• To initiate the study of $k$-Colouring for $(P_r, \overline{P_r})$-free graphs, particularly for $r=6$.
• To prove NP-completeness of 5-Colouring for $(P_6, \overline{P_1 + P_6})$-free graphs using a reduction from 3-SAT via a constructed graph $G_{H,I}$.

Proposed method

• Leveraged complement-closure of $(H, \overline{H})$-free graph classes to equate the complexity of Colouring and Clique Cover.
• Combined known results on $H$-free graphs with new constructions to classify complexity for $(H, \overline{H})$-free graphs.
• Used a reduction from 3-SAT to prove NP-completeness of 5-Colouring, constructing a graph $G_{H,I}$ based on a 4-critical $P_6$-free graph $H$.
• Employed structural analysis of induced subgraphs to show that $G_{H,I}$ avoids $P_6$ and $\overline{P_1 + P_6}$, ensuring the reduction is valid.
• Analyzed vertex types (C, D, X, U) in the construction to rule out induced $\overline{P_1 + P_6}$ subgraphs via case analysis on possible vertex sets.
• Applied Ramsey-theoretic reasoning and induced subgraph enumeration to derive contradictions when assuming the existence of forbidden subgraphs.

Experimental results

Research questions

• RQ1 For which graphs $H$ is Colouring polynomial-time solvable on $(H, \overline{H})$-free graphs?
• RQ2 What is the complexity of $k$-Colouring for $(P_r, \overline{P_r})$-free graphs, particularly for $r=6$?
• RQ3 Which $(H_1, H_2)$-free graph classes with $H_2 = \overline{H_1}$ remain open after known classifications?
• RQ4 Can the complexity of Colouring be fully classified for graph classes closed under complementation?
• RQ5 Is 5-Colouring NP-complete for $(P_6, \overline{P_1 + P_6})$-free graphs?

Key findings

• The complexity of Colouring and Clique Cover coincide for $(H, \overline{H})$-free graphs due to complement-closure.
• Colouring is polynomial-time solvable for $(H, \overline{H})$-free graphs if $H \subseteq_i P_4$; otherwise, it is NP-complete.
• The only unresolved cases for $(H, \overline{H})$-free graphs are $H = sP_1 + P_3$ for $s \geq 3$ and $H = sP_1 + P_4$ for $s \geq 2$.
• 5-Colouring is NP-complete for $(P_6, \overline{P_1 + P_6})$-free graphs, proven via a reduction from 3-SAT using a $P_6$-free 4-critical graph.
• The constructed graph $G_{H,I}$ is both $P_6$-free and $\overline{P_1 + P_6}$-free, validating the NP-completeness reduction.
• Structural analysis of vertex types and induced subgraphs ruled out the existence of $\overline{P_1 + P_6}$ in $G_{H,I}$, completing the contradiction proof.