Snakes and Ladders: a Treewidth Story

TL;DR

This work analyzes how ladder-like substructures affect the treewidth of graphs and shows that long ladders can be shortened without increasing or decreasing tw under precise conditions. Specifically, for graphs with $tw(G) \ge 4$, ladders of length at least 3 can be extended indefinitely without increasing tw, and all ladders can be shortened to length 4 without decreasing tw (with tightness proven). The paper also connects these graph-theoretic results to phylogenetics by proving that common chain reductions in display graphs preserve treewidth, with chains reducible to four leaves (and not safely to three) in general, while display-graph structure permits stronger bounds. These findings yield a safe, universal ladder-length reduction rule, inform minimal forbidden minor characterizations for bounded treewidth, and offer constructive approaches for treewidth-preserving reductions relevant to phylogenetic analyses and beyond.

Abstract

Let $G$ be an undirected graph. We say that $G$ contains a ladder of length $k$ if the $2 \times (k+1)$ grid graph is an induced subgraph of $G$ that is only connected to the rest of $G$ via its four cornerpoints. We prove that if all the ladders contained in $G$ are reduced to length 4, the treewidth remains unchanged (and that this bound is tight). Our result indicates that, when computing the treewidth of a graph, long ladders can simply be reduced, and that minimal forbidden minors for bounded treewidth graphs cannot contain long ladders. Our result also settles an open problem from algorithmic phylogenetics: the common chain reduction rule, used to simplify the comparison of two evolutionary trees, is treewidth-preserving in the display graph of the two trees.

Figures (9)

• Figure 1: A ladder $L$ of length 3 with corner points $a,b,c,d$.
• Figure 2: Inserting a new edge $\{u',v'\}$ into ladder $L$ results in ladder $L'$ of length 4.
• Figure 3: Path $p_{ab}$ goes via the interior, but it cannot be relabelled to $b$ because it is used by other paths $p_{az}$ to some neighbour $z$ of $a$ that does not lie on the ladder.
• Figure 4: The bags $B'$, $B_{v\text{-done}}$ and $B_{a\text{-done}}$ illustrated. Note that $B_{a\text{-done}}$ cannot be the penultimate bag on the path $P$ from $B_1$ to $B_2$, due to the presence of $d$ in that bag.
• Figure 5: A graph of treewidth 3 that contains a ladder with 3 squares, shown in red. Increasing the length of the ladder by 1 square increases the treewidth to 4.

Theorems & Definitions (24)

• lemma thmcounterlemma: kelk2017treewidth
• lemma thmcounterlemma: kelk2017treewidth
• proof
• lemma thmcounterlemma
• proof
• theorem thmcountertheorem: kelk2017treewidth
• theorem thmcountertheorem
• proof
• proof
• proof