Colouring Graphs Without a Subdivided H-Graph: A Full Complexity Classification

TL;DR

This work studies the Colouring problem on $H$-subgraph-free graphs, focusing on subdivided $H$-graphs and providing a near-complete complexity classification. It introduces structural tools—$T$-type/$L$-type subgraphs, jumps, and chain extensions—to decompose graphs and bound treedepth in large components, enabling efficient colouring. The authors achieve a full polynomial-time classification for subdivided $\mathbb{H}_0$ and subdivided $\mathbb{H}_1$, leaving only two open connected-$H$ families, and extend the techniques to the Stable Cut problem. These results significantly advance our understanding of colouring in monotone, subgraph-free graph classes and demonstrate versatile structural methods applicable to related problems.

Abstract

We consider Colouring on graphs that are $H$-subgraph-free for some fixed graph $H$, i.e., graphs that do not contain $H$ as a subgraph. It is known that even $3$-Colouring is NP-complete for $H$-subgraph-free graphs whenever $H$ has a cycle; or a vertex of degree at least $5$; or a component with two vertices of degree $4$, while Colouring is polynomial-time solvable for $H$-subgraph-free graphs if $H$ is a forest of maximum degree at most $3$, in which each component has at most one vertex of degree $3$. For connected graphs $H$, this means that it remains to consider when $H$ is tree of maximum degree $4$ with exactly one vertex of degree $4$, or a tree of maximum degree $3$ with at least two vertices of degree $3$. We let $H$ be a so-called subdivided "H"-graph, which is either a subdivided $\mathbb{H}_0$: a tree of maximum degree $4$ with exactly one vertex of degree $4$ and no vertices of degree $3$, or a subdivided $\mathbb{H}_1$: a tree of maximum degree $3$ with exactly two vertices of degree $3$. In the literature, only a limited number of polynomial-time and NP-completeness results for these cases are known. We develop new polynomial-time techniques that allow us to determine the complexity of Colouring on $H$-subgraph-free graphs for all the remaining subdivided "H"-graphs, so we fully classify both cases. As a consequence, the complexity of Colouring on $H$-subgraph-free graphs has now been settled for all connected graphs $H$ except when $H$ is a tree of maximum degree $4$ with exactly one vertex of degree $4$ and at least one vertex of degree $3$; or a tree of maximum degree $3$ with at least three vertices of degree $3$. We also employ our new techniques to obtain the same new polynomial-time results for another classic graph problem, namely Stable Cut.

Figures (6)

• Figure 1: The trees $T_1,\ldots, T_6$ from GPR15; note that $T_1=\mathbb{H}_1^{2,2,2,2}$ and $T_3=S_{2,2,2,2}$.
• Figure 2: The "H"-graph $\mathbb{H}=\mathbb{H}_1=\mathbb{H}_1^{1,1,1,1}$ and for $i\geq 1$, the graph $\mathbb{H}_i=\mathbb{H}_i^{1,1,1,1}$ obtained from $\mathbb{H}$ by subdividing $i-1$ times the horizontal edge $uv$. The other edges of $\mathbb{H}$ are the vertical edges.
• Figure 3: An illustration of the notation regarding paths. The subpath $P[p_1:p_2] = P[1:2]$ is highlighted by the purple box, $P[p_4:p_7] = P[4:7]$ is highlighted by the teal box and $P[p_8:p_{10}] = P[8:10]$ is highlighted by the orange box. Note that negative indexing may also be used i.e., $P[8:-1] = P[-3:-1] = P[8:10]$.
• Figure 4: The fan graph $F_{10}$ with poles drawn in orange.
• Figure 5: An illustration of Definitions \ref{['def-Z^P(u,v)']}, \ref{['def-jump-out']}, and \ref{['def-jump-out-max']}. The jump $Z^P(p_6,p_{10})$ is highlighted in teal and the jump $Z^P(p_5,p_9)$ in orange. There is a negative jump out of $(p_4,p_7)$. There are three positive jumps out of $(p_4,p_7)$ and these have endpoints $(p_5,p_{9})$, $(p_5,p_{10})$ and $(p_6,p_{10})$, respectively. Here the maximum positive jump out of $(p_4,p_7)$ has endpoints $(p_6,p_{10})$, that is $(x^+(p_4,p_7), y^+(p_4,p_7)) = (p_6,p_{10})$.

Theorems & Definitions (17)

• Theorem 1
• Theorem 2
• Theorem 3: NO12
• Theorem 4: GALVIN19827
• Definition 5: $T$-type subgraph
• Definition 6: $L$-type subgraph
• Definition 7: Minimal $T$-type ($L$-type) subgraph
• Lemma 8
• Definition 9: Jump
• Definition 10: Positive/negative jump out of an interval