On algorithmic applications of sim-width and mim-width of $(H_1, H_2)$-free graphs

TL;DR

The paper investigates algorithmic consequences of sim-width and mim-width for hereditary graph classes, with a focus on $(H_1,H_2)$-free graphs. It proves List $k$-Colouring is XP for graph classes with sim-width bounded and quickly computable, and shows that if Independent Set is polynomial-time on such classes then Maximum Weight Independent $\mathcal{H}$-Packing is also polynomial-time; it also advances mim-width classifications for several $(H_1,H_2)$-free families. A key contribution is a complete mim-width dichotomy for $(rP_1, \overline{K_{t,s}+P_1})$-free graphs and substantial progress for $(K_r,H)$-free graphs with small $H$, together with a sim-width preserving $\mathcal{H}$-graph framework that facilitates transferring complexity results. The work highlights how bounded sim-width can yield tractable algorithms via XP frameworks and branch decompositions, while delineating open problems, notably whether Independent Set remains polynomial-time on sim-width-bounded classes and the remaining mim-width border cases.

Abstract

Mim-width and sim-width are among the most powerful graph width parameters, with sim-width more powerful than mim-width, which is in turn more powerful than clique-width. While several $\mathsf{NP}$-hard graph problems become tractable for graph classes whose mim-width is bounded and quickly computable, no algorithmic applications of boundedness of sim-width are known. In [Kang et al., A width parameter useful for chordal and co-comparability graphs, Theoretical Computer Science, 704:1-17, 2017], it is asked whether \textsc{Independent Set} and \textsc{$3$-Colouring} are $\mathsf{NP}$-complete on graphs of sim-width at most $1$. We observe that, for each $k \in \mathbb{N}$, \textsc{List $k$-Colouring} is polynomial-time solvable for graph classes whose sim-width is bounded and quickly computable. Moreover, we show that if the same holds for \textsc{Independent Set}, then \textsc{Independent $\mathcal{H}$-Packing} is polynomial-time solvable for graph classes whose sim-width is bounded and quickly computable. This problem is a common generalisation of \textsc{Independent Set}, \textsc{Induced Matching}, \textsc{Dissociation Set} and \textsc{$k$-Separator}. We also make progress toward classifying the mim-width of $(H_1,H_2)$-free graphs in the case $H_1$ is complete or edgeless. Our results solve some open problems in [Brettell et al., Bounding the mim-width of hereditary graph classes, Journal of Graph Theory, 99(1):117-151, 2022].

Figures (6)

• Figure 1: The graph $\overline{K_{4,4}+P_1}$.
• Figure 2: How to construct a branch decomposition $(T', \delta')$ of $\mathcal{H}(G)$ from a branch decomposition $(T, \delta)$ of $G$. We distinguish vertices $t_i$ such that $|F(v_{t_i})| = 0$ ($i = 2$), $|F(v_{t_i})| = 1$ ($i = 3$) and $|F(v_{t_i})| \geq 2$ ($i = 1$).
• Figure 3: How to construct a branch decomposition $(T', \delta')$ of $G'$ from a branch decomposition $(T, \delta)$ of $G$, where $G'$ is obtained from $G$ by adding a leaf vertex $u$ adjacent to $v$.
• Figure 4: The different colourings of elementary walls used in the proofs of \ref{['344t1', '433t1', '522t1']}.
• Figure 5: The graph $W_4$ with the red-blue colouring as in the proof of \ref{['K5unbound']}.

Theorems & Definitions (78)

• Theorem 1: Kw20
• Theorem 2: BHMP22
• Proposition 3: see Proof of Proposition 4.2 in KKST17
• Theorem 4
• proof
• Lemma 5
• proof : Proof sketch
• Theorem 6
• Corollary 6
• Theorem 7