Maximum $k$-colourable induced subgraphs in $(P_5+rK_1)$-free graphs

TL;DR

The paper proves that for fixed $k$ and $r$, the Weighted Maximum List-$k$-Colourable Induced Subgraph problem (WML$k$CIS) is solvable in polynomial time on $(P_5+rK_1)$-free graphs with arbitrary list-$k$-assignments and weights. It achieves this by introducing a canvas-based framework that reduces the problem to a polynomial-sized family of connected $L$-colourable subgraphs and then to a maximum-weight independent set problem on a blob graph $H(G,\mathcal{C})$, with an inductive solution on $k$ that uses WM$(k-1)$CIS as a subroutine. Key technical tools include edge-contraction invariance for $(P_5+rK_1)$-free graphs, the blob graph construction, and Camby–Schaudt-type domination bounds to limit candidate subgraphs. Together, these enable a polynomial-time algorithm and contribute to the growing dichotomy for WML$k$CIS in $H$-free graphs, placing $(P_5+rK_1)$-free graphs among the tractable cases for large $k$.

Abstract

We show that for any nonnegative integer $r$, the Weighted Maximum List-$k$-Colourable Induced Subgraph problem can be solved in polynomial time for input graphs that do not contain $(P_5+ rK_1)$ as an induced subgraph, and give an explicit algorithm demonstrating this. This answers a question of Agrawal et al.\ (2024).

Figures (2)

• Figure 1: The structure described in Claim \ref{['claim:aissmall']}. Here, $X = N(A'_c) \cap (M\setminus W).$ Dashed lines indicate non-adjacency. The path $u_1Pu_2$ is the shortest $(u_1,u_2)$-path with internal vertices in $S$, and hence if $s_1 \neq s_2$, we have that $u_2s_1$ and $u_1s_2$ are not edges in $G$. To keep the image uncluttered, the dashed lines between $S$ and each of $X$ and $B_j$ as well as those between $A'_c$ and $B_j$ have been omitted. On the left, $m(u_1),u_1,P,u_2,m(u_2)$ together with $B_j$ contains an induced copy of $(P_5 + rK_1)$. On the right, $u_1,P,u_2,m(u_2),m(u_3)$ with $B_j$ contains an induced copy of $(P_5 + rK_1)$. As both cases contradict that $G \in \mathcal{G}_r$, we conclude that $|A_c| \leq 2k$.
• Figure 2: The structure described in Lemma \ref{['lem:solwithallinC']}, Claim \ref{["claim:Y'deffocanvas"]} with (for illustrative purposes) $\ell = 2$ and $i = 1$. Dashed lines indicate non-adjacency. To keep the image uncluttered, the dashed lines between $Z_{2,c}$ and each of $\{y_1,y_2,y_3\}$ and $\{n_2(y_1), n_2(y_2), n_2(y_3)\}$ have been omitted. On the left: the first case covered in Claim \ref{["claim:Y'deffocanvas"]}, where $n_2(y_1)$ and $n_2(y_2)$ are non-adjacent: here $n_2(y_1),y_1,v_1,y_2,n_2(y_2)$ and $Z_{2,c}$ form an induced copy of $(P_5+rK_1)$ shown. On the right, the second case covered in Claim \ref{["claim:Y'deffocanvas"]}: here, we assume $\{n_2(y_1),n_2(y_2),n_2(y_3)\}$ form a clique, and so $n_2(y_3),n_2(y_1),y_1,v_1,y_2$ and $Z_{2,c}$ form an induced copy of $(P_5 + rK_1)$. Since both cases lead to a contradiction, we conclude that $|Y_{1,c,2}| \leq 2$.

Theorems & Definitions (40)

• Theorem 1
• Theorem 2: galby2025, Theorem 4
• Lemma 3
• proof
• Lemma 4
• proof
• Definition 1
• Lemma 5
• proof
• Lemma 6