Four NP-complete problems about generalizations of perfect graphs

TL;DR

This paper proves four problems related to perfect graphs are NP-complete, expanding understanding of computational complexity in graph theory and perfect graph generalizations.

Contribution

It establishes NP-completeness for four new problems involving partitions and properties of perfect graphs, using reductions from 3- and 4-colorability of triangle-free graphs.

Findings

Four NP-complete problems about perfect graph generalizations.

NP-completeness holds even for triangle-free graphs.

Uses reductions from known coloring problems.

Abstract

We show that the following problems are NP-complete. 1. Can the vertex set of a graph be partitioned into two sets such that each set induces a perfect graph? 2. Is the difference between the chromatic number and clique number at most $1$ for every induced subgraph of a graph? 3. Can the vertex set of every induced subgraph of a graph be partitioned into two sets such that the first set induces a perfect graph, and the clique number of the graph induced by the second set is smaller than that of the original induced subgraph? 4. Does a graph contain a stable set whose deletion results in a perfect graph? The proofs of the NP-completeness of the four problems follow the same pattern: Showing that all the four problems are NP-complete when restricted to triangle-free graphs by using results of Maffray and Preissmann on $3$-colorability and $4$-colorability of triangle-free graphs

Full text

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Can the vertex set of a graph be partitioned into two sets such that each set induces a perfect graph?

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Is the difference between the chromatic number and clique number at most $1$ for every induced subgraph of a graph?

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Can the vertex set of every induced subgraph of a graph be partitioned into two sets such that the first set induces a perfect graph, and the clique number of the graph induced by the second set is smaller than that of the original induced subgraph?

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Does a graph contain a stable set whose deletion results in a perfect graph?

The proofs of the NP-completeness of the four problems follow the same pattern: Showing that all the four problems are NP-complete when restricted to triangle-free graphs by using results of Maffray and Preissmann [3] on $3$-colorability and $4$-colorability of triangle-free graphs.

1 Introduction

All graphs considered in this article are finite and simple. Let $G$ be a graph. The complement $G^{c}$ of $G$ is the graph with vertex set $V(G)$ and such that two vertices are adjacent in $G^{c}$ if and only if they are non-adjacent in $G$. For two graphs $H$ and $G$, $H$ is an induced subgraph of $G$ if $V(H)\subseteq V(G)$, and a pair of vertices $u,v\in V(H)$ is adjacent if and only if it is adjacent in $G$. We say that $G$ contains $H$ if $G$ has an induced subgraph isomorphic to $H$. If $G$ does not contain $H$, we say that $G$ is $H$-free. For a set $X\subseteq V(G)$ we denote by $G[X]$ the induced subgraph of $G$ with vertex set $X$. A hole in a graph is an induced subgraph that is isomorphic to the cycle $C_{k}$ with $k\geq 4$, and $k$ is the length of the hole. A hole is odd if $k$ is odd, and even otherwise. The chromatic number of a graph $G$ is denoted by $\chi(G)$ and the clique number by $\omega(G)$. $G$ is called perfect if for every induced subgraph $H$ of $G$, $\chi(H)=\omega(H)$. $G$ is said to be perfectly divisible if for all induced subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A,B$ such that $H[A]$ is perfect and $\omega(B)<\omega(H)$. $G$ is said to be nice if for every induced subgraph $H$ of $G$, $\chi(H)-\omega(H)\in\{0,1\}$. $G$ is said to be 2-perfect if $V(G)$ can be partitioned into two sets $A,B$ such that both $G[A]$ and $G[B]$ are perfect. $G$ is said to be stable-perfect if $G$ contains a stable set $S$ such that $G\setminus S$ is perfect. Note that perfect graphs are stable-perfect, and stable-perfect graphs are 2-perfect, perfectly divisible, and nice. In this note, we show that the recognition problems for the four classes ($2$-perfect, nice, perfectly divisible, stable-perfect) are NP-complete, a stark contrast to the existence of a polynomial-time recognition algorithm for perfect graphs [1].

2 Four NP-complete problems

We need the following results from [3].

Theorem 2.1** (Maffray-Preissmann).**

It is NP-complete to determine whether a triangle-free graph is $3$-colorable.

Theorem 2.2** (Maffray-Preissmann).**

It is NP-complete to determine whether a triangle-free graph is $4$-colorable.

The following is a basic fact about perfect graphs.

Lemma 2.1**.**

A triangle-free graph is perfect if and only if it is bipartite.

Proof.

Since bipartite graphs are perfect, one direction is trivial. To prove the other direction, let $G$ be a triangle-free perfect graph. Since $G$ contains neither a triangle nor an odd hole, it contains no odd cycle as a subgraph. Hence $G$ is bipartite. ∎

We first prove the NP-completeness of recognizing $2$-perfect graphs. First we need a lemma.

Lemma 2.2**.**

A triangle-free graph is 2-perfect if and only if it is $4$-colorable.

Proof.

This follows easily from Lemma 2.1. ∎

Theorem 2.3**.**

It is NP-complete to determine whether a graph is $2$-perfect.

Proof.

We show that the restricted problem of determining whether a triangle-free graph is $2$-perfect is NP-complete. Let $G$ be a triangle-free graph. By Lemma 2.2, $G$ is $2$-perfect if and only if $G$ is $4$-colorable. By Theorem 2.2 it is NP-complete to determine whether a triangle-free graph is $4$-colorable, We thus conclude that it is NP-complete to determine whether a triangle-free graph is $2$-perfect. ∎

We now move on to the classes of perfectly divisible graphs, stable-perfect, and nice graphs. Problem 32 in [4] asks whether nice graphs can be recognized in polynomial time. The recognition problem for nice graphs turns out to be NP-complete. The following lemma tells that for triangle-free graphs, the three classes mentioned above are equivalent to the class of $3$-colorable graphs.

Lemma 2.3**.**

For a triangle-free graph $G$, the following are equivalent:

1. (i)

$G$ is $3$-colorable. 2. (ii)

$G$ is perfectly divisible. 3. (iii)

$G$ is stable-perfect. 4. (iv)

$G$ is nice.

Proof.

We prove the following chain of implications $(i)\Rightarrow(ii)\Rightarrow(iii)\Rightarrow(iv)\Rightarrow(i)$.

$(i)\Rightarrow(ii)$: Suppose $G$ is $3$-colorable. Let $H$ be an induced subgraph of $G$. Note that $H$ is also $3$-colorable. We may assume that $H$ has clique number $2$. Let $(S_{1},S_{2},S_{3})$ be a partition of $V(H)$ into three stable sets. Now $(S_{1}\cup S_{2},S_{3})$ is a partition of $V(G)$ as in the definition of being perfectly divisible. We conclude that $G$ is perfectly divisible.

$(ii)\Rightarrow(iii)$: Suppose $G$ is perfectly divisible. Hence there is a partition of $V(G)$ into sets $A,B$ such that $G[A]$ is perfect and $\omega(B)<\omega(G)$. Since $G$ has no triangles, this implies that $B$ is a stable set. Thus $G$ is stable-perfect.

$(iii)\Rightarrow(iv)$: Suppose $G$ is stable-perfect. Let $H$ be an induced subgraph of $G$. We may assume that $H$ has clique number $2$. Thus $H$ contains a stable set $S$ such that $H\setminus S$ is perfect. Since $H$ is also triangle-free, by Lemma 2.1, $H\setminus S$ is bipartite. Hence the chromatic number of $H$ is at most $3$. We conclude that $G$ is nice.

$(iv)\Rightarrow(i)$: Suppose $G$ is nice. Since $G$ is triangle-free, its clique number is at most $2$. Since $G$ is nice, we conclude that its chromatic number is at most $3$. Thus $G$ is $3$-colorable.

This concludes the proof of all the implications, and proves the theorem. ∎

Theorem 2.4**.**

The following problems are NP-complete:

Given a graph, is it perfectly divisible? 2. 2.

Given a graph, is it stable-perfect? 3. 3.

Given a graph, is it nice?

Proof.

By Lemma 2.3 and Theorem 2.1, the problems are already NP-complete when restricted to triangle-free graphs. ∎

3 Open problems

$G$ is said to be $2$-divisible if for all induced subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A,B$ such that $\omega(A)<\omega(H)$ and $\omega(B)<\omega(H)$.

Conjecture 3.1**.**

It is NP-complete to determine whether a graph is $2$-divisible.

There is a nice conjecture about $2$-divisible graphs:

Conjecture 3.2** (Hoang-McDiarmid [2]).**

A graph is $2$-divisible if and only if it is odd-hole-free.

The complexity of the recognition of odd-hole-free graphs is also unknown.

Conjecture 3.3**.**

It is NP-complete to determine whether a graph contains an odd hole.

4 Acknowledgment

I would like to thank Maria Chudnovsky for some useful discussion about perfectly divisible graphs which inspired this note.