Solving Problems on Generalized Convex Graphs via Mim-Width

TL;DR

The paper studies generalized ${\cal H}$-convex bipartite graphs and uses mim-width, along with refined width parameters, to explain when key NP-hard problems become tractable. It proves bounded mim-width for circular convex graphs and for $(t,\Delta)$-tree convex graphs with $t\ge 1$, $\Delta\ge 3$, and unbounded mim-width for star convex and comb convex graphs, establishing a clear width-based boundary. It then derives algorithmic consequences for Locally Checkable Vertex Subset problems and List $k$-Colouring, yielding polynomial-time solvability in the bounded-width cases and NP-completeness in the unbounded cases, thereby providing dichotomies for several problems. A refined analysis of width parameters reveals how width notions relate across these classes, showing that bounded mim-width does not imply bounded thinness or clique-width and detailing when linear mim-width or sim-width provide stronger bounds. The work thus unifies and extends existing results by linking reductions to convex graphs with a width-parameter framework, offering a systematic path to tractability results across broad graph classes.

Abstract

A bipartite graph $G=(A,B,E)$ is ${\cal H}$-convex, for some family of graphs ${\cal H}$, if there exists a graph $H\in {\cal H}$ with $V(H)=A$ such that the set of neighbours in $A$ of each $b\in B$ induces a connected subgraph of $H$. Many $\mathsf{NP}$-complete problems, including problems such as Dominating Set, Feedback Vertex Set, Induced Matching and List $k$-Colouring, become polynomial-time solvable for ${\mathcal H}$-convex graphs when ${\mathcal H}$ is the set of paths. In this case, the class of ${\mathcal H}$-convex graphs is known as the class of convex graphs. The underlying reason is that the class of convex graphs has bounded mim-width. We extend the latter result to families of ${\mathcal H}$-convex graphs where (i) ${\mathcal H}$ is the set of cycles, or (ii) ${\mathcal H}$ is the set of trees with bounded maximum degree and a bounded number of vertices of degree at least $3$. As a consequence, we can re-prove and strengthen a large number of results on generalized convex graphs known in the literature. To complement result (ii), we show that the mim-width of ${\mathcal H}$-convex graphs is unbounded if ${\mathcal H}$ is the set of trees with arbitrarily large maximum degree or an arbitrarily large number of vertices of degree at least $3$. In this way we are able to determine complexity dichotomies for the aforementioned graph problems. Afterwards we perform a more refined width-parameter analysis, which shows even more clearly which width parameters are bounded for classes of ${\cal H}$-convex graphs.

Figures (5)

• Figure 1: (\ref{['fig:circ']}) A circular convex graph $G = (A, B, E)$ with a circular ordering on $A$. (\ref{['fig:branch']}) A (linear) branch decomposition $(T, \delta)$ for $G$, where $T$ is a caterpillar with a specified edge $e$, together with the graph $G[A_e, \overline{A_e}]$. The bold edges in $G[A_e, \overline{A_e}]$ form an induced matching and it is easy to see that $\mathrm{cutmim}_{G}(A_{e}, \overline{A_{e}}) = 2$.
• Figure 2: The inclusion relations between the classes we consider. A line from a lower-level class to a higher one means the first class is contained in the second.
• Figure 3: A $3$-thin representation of a graph. The vertices are ordered increasingly by their $y$-coordinate, and the classes correspond to the vertical lines.
• Figure 4: The relationships between the different width parameters that we consider in Section \ref{['s-refined']}. Parameter $p$ is more powerful than parameter $q$ if and only if there exists a directed path from $p$ to $q$. To explain the incomparabilities, proper interval graphs have proper thinness 1 BE19 and unbounded clique-width GR00, whereas trees have tree-width 1 and unbounded linear mim-width HTV19. Unreferenced arrows follow from the definitions of the width parameters involved except for the arrow from proper thinness to path-width whose proof we give in Section \ref{['s-new']}.
• Figure 6: The graphs $G_1$, $G_2$, $G_3$, from the family of graphs $\{G_k\}_{k\geq 1}$ in the proof of Theorem \ref{['proper']}. All the graphs in this family are convex and their proper thinness increases with $k$.

Theorems & Definitions (32)

• theorem thmcountertheorem
• theorem thmcountertheorem
• theorem thmcountertheorem
• corollary thmcountercorollary
• corollary thmcountercorollary
• theorem thmcountertheorem
• corollary thmcountercorollary
• lemma thmcounterlemma: see, e.g., Buchin et al. BKM11
• theorem thmcountertheorem
• proof