Maximum list $r$-colorable induced subgraphs in $kP_3$-free graphs

TL;DR

The authors address the Max-Weight List $r$-Colorable Induced Subgraph problem, a broad generalization of problems such as Max-Weight Independent Set and Odd Cycle Transversal, restricted to $kP_3$-free graphs. They introduce amiable and distance-$d$ amiable families to capture a polynomial-sized search space that contains all maximal independent sets or distance-$d$ independent sets. For fixed $r$ and $k$, they present a polynomial-time algorithm that reduces Max-Weight List $r$-Colorable Induced Subgraph to a collection of weighted bipartite-matching subproblems over an amiable family, yielding tractability results and implications for Odd Cycle Transversal and List $r$-Coloring on $H$-free graphs. They also prove polynomial-time algorithms for distance-$d$ generalizations with $d \ge 6$, shedding light on the computational landscape and outlining remaining open cases for $d \in \{3,4,5\}$ and certain $H$-free configurations.

Abstract

We show that, for every fixed positive integers $r$ and $k$, \textsc{Max-Weight List $r$-Colorable Induced Subgraph} admits a polynomial-time algorithm on $kP_3$-free graphs. This problem is a common generalization of \textsc{Max-Weight Independent Set}, \textsc{Odd Cycle Transversal} and \textsc{List $r$-Coloring}, among others. Our result has several consequences. First, it implies that, for every fixed $r \geq 5$, assuming $\mathsf{P}\neq \mathsf{NP}$, \textsc{Max-Weight List $r$-Colorable Induced Subgraph} is polynomial-time solvable on $H$-free graphs if and only if $H$ is an induced subgraph of either $kP_3$ or $P_5+kP_1$, for some $k \geq 1$. Second, it makes considerable progress toward a complexity dichotomy for \textsc{Odd Cycle Transversal} on $H$-free graphs, allowing to answer a question of Agrawal, Lima, Lokshtanov, Rz{ą}{ż}ewski, Saurabh, and Sharma [TALG 2024]. Third, it gives a short and self-contained proof of the known result of Chudnovsky, Hajebi, and Spirkl [Combinatorica 2024] that \textsc{List $r$-Coloring} on $kP_3$-free graphs is polynomial-time solvable for every fixed $r$ and $k$. We also consider two natural distance-$d$ generalizations of \textsc{Max-Weight Independent Set} and \textsc{List $r$-Coloring} and provide polynomial-time algorithms on $kP_3$-free graphs for every fixed integers $r$, $k$, and $d \geq 6$.

Figures (2)

• Figure 1: Visualization for distance-$d$ amiable family $\mathcal{S} = \{S_1, S_2, S_3\}$. Circles represent cliques and $\alpha$, $\beta$ are maximal distance-$d$ independent sets. Dashed lines depict paths of lengths at least $d$.
• Figure 2: The case $d \leq 5$ (paths in blue are of length $d-3$, those in red, $d-2$).

Theorems & Definitions (23)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Lemma 6
• Lemma 7
• proof
• proof : Proof of \ref{['amiablefam:P3-freeness']}