Complexity Framework for Forbidden Subgraphs II: Edge Subdivision and the "H"-graphs

TL;DR

This work investigates the complexity of four classic problems on $\mathcal{H}$-subgraph-free graphs where $\mathcal{H}$ is drawn from the infinite family of subdivided 'H'-graphs $\{\mathbb{H}_i: i\ge1\}$. It reveals a nuanced landscape where the boundary between polynomial-time solvability and NP-hardness varies across problems when certain $\mathbb{H}_i$-subdivisions are forbidden, and where some problems admit polynomial-time certifying algorithms on these restricted classes (e.g., $C_5$-Colouring on $\mathbb{H}_3$-subgraph-free graphs and Star $3$-Colouring on particular $\mathbb{H}_2$-free families). The authors develop algorithmic techniques, including structural reductions (e.g., the diamond and butterfly reductions) and finite forbidden-subgraph certificates, to obtain polynomial-time and certifying results, while also proving NP-hardness for other settings (e.g., $k$-Induced Disjoint Paths on subcubic graphs forbidding $\mathbb{H}_{4},\ldots,\mathbb{H}_{\ell}$). The findings demonstrate a rich, problem-specific complexity landscape within the $C12$-framework and motivate several open questions about complete dichotomies and extensions to related problems.

Abstract

For a fixed set ${\cal H}$ of graphs, a graph $G$ is ${\cal H}$-subgraph-free if $G$ does not contain any $H \in {\cal H}$ as a (not necessarily induced) subgraph. A recently proposed framework gives a complete classification on ${\cal H}$-subgraph-free graphs (for finite sets ${\cal H}$) for problems that are solvable in polynomial time on graph classes of bounded treewidth, NP-complete on subcubic graphs, and whose NP-hardness is preserved under edge subdivision. While a lot of problems satisfy these conditions, there are also many problems that do not satisfy all three conditions and for which the complexity in ${\cal H}$-subgraph-free graphs is unknown. We study problems for which only the first two conditions of the framework hold (they are solvable in polynomial time on classes of bounded treewidth and NP-complete on subcubic graphs, but NP-hardness is not preserved under edge subdivision). In particular, we make inroads into the classification of the complexity of four such problems: Hamilton Cycle, $k$-Induced Disjoint Paths, $C_5$-Colouring and Star $3$-Colouring. Although we do not complete the classifications, we show that the boundary between polynomial time and NP-complete differs among our problems and also from problems that do satisfy all three conditions of the framework, in particular when we forbid certain subdivisions of the ``H''-graph (the graph that looks like the letter ``H''). Hence, we exhibit a rich complexity landscape among problems for ${\cal H}$-subgraph-free graph classes.

Figures (17)

• Figure 1: BJMOPPSV Left: A graph in ${\cal S}$: the graph $S_{3,3,3}+P_2+P_3+P_4$; note that $S_{3,3,3}$ is the $2$-subdivision of the claw $K_{1,3}$. Right: the graphs $\mathbb{H}_1$ and $\mathbb{H}_3$; here, $\mathbb{H}_1$ is the "H"-graph, formed by an edge (the middle edge) joining the middle vertices of two $P_3$s, and $\mathbb{H}_i$ ($i\geq 2$) is obtained from $\mathbb{H}_1$ by $(i-1)$-subdividing the middle edge.
• Figure 2: The tree $T$.
• Figure 5: Rule 1. Possible connections in our subgraph (left). What we replace this subgraph with (right). Dotted lines are possible additional edges.
• Figure 6: Rule 2. Possible connections in our subgraph (left). What we replace this subgraph with (right).
• Figure 7: A (partial) $C_5$-flower and the three exceptional $\mathbb{H}_3$-subgraph-free $C_5$-critical graphs $E_1$, $E_2$ and $E_3$.

Theorems & Definitions (44)

• Theorem 1: JMOPPSV
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 13: $\spadesuit$
• Theorem 14: $\spadesuit$
• Theorem 15: $\spadesuit$
• Lemma 16