Graph parameters that are coarsely equivalent to path-length

TL;DR

The paper studies graph parameters that are coarsely equivalent to path-length, establishing a network of constants-approximate relations among $pl(G)$ and several new or existing measures. It introduces $adc(G)$ (distance-$k$-approximating caterpillars), the McCarty index $mci(G)$, the fatness index $mfi(G)$, and the power-AT/cocomparability framework via $pat(G)$ and $pcc(G)$, showing these are within constant factors of $pl(G)$. Key results connect $pl(G)$ to quasi-isometries with paths and to embeddings into caterpillars, AT-free graphs, and cocomparability graphs, yielding both structural characterizations and algorithmic consequences (e.g., $O(n^3)$-time approximations for $adc(G)$). The work also provides alternative proofs and bounds for characterizations of bounded path-length in terms of forbidden $K$-fat $K_3$- and $K_{1,3}$-minors, tying together domination notions such as $k$-dominating paths and pairs. Overall, the paper unifies multiple graph-structural parameters under the path-length lens, offering both theoretical insight and potential algorithmic applications for bounded-path-length graph classes.

Abstract

Two graph parameters are said to be coarsely equivalent if they are within constant factors from each other for every graph $G$. Recently, several graph parameters were shown to be coarsely equivalent to tree-length. Recall that the length of a tree-decomposition ${\cal T}(G)$ of a graph $G$ is the largest diameter of a bag in ${\cal T}(G)$, and the tree-length $tl(G)$ of $G$ is the minimum of the length, over all tree-decompositions of $G$. Similarly, the length of a path-decomposition ${\cal P}(G)$ of a graph $G$ is the largest diameter of a bag in ${\cal P}(G)$, and the path-length $pl(G)$ of $G$ is the minimum of the length, over all path-decompositions of $G$. In this paper, we present several graph parameters that are coarsely equivalent to path-length. Among other results, we show that the path-length of a graph $G$ is small if and only if one of the following equivalent conditions is true: (a) $G$ can be embedded to an unweighted caterpillar tree (equivalently, to a graph of path-width one) with a small additive distortion; (b) there is a constant $r\ge 0$ such that for every triple of vertices $u,v,w$ of $G$, disk of radius $r$ centered at one of them intercepts all paths connecting two others; (c) $G$ has a $k$-dominating shortest path with small $k\ge 0$; (d) $G$ has a $k'$-dominating pair with small $k'\ge 0$; (e) some power $G^μ$ of $G$ is an AT-free (or even a cocomparability) graph for a small integer $μ\ge 0$.

Figures (3)

• Figure 1: Illustration to the proof of Theorem \ref{['th:pat-pl']} (part for ${\sf dpr}(G)\le 3\cdot{\sf dsp}(G)$).
• Figure 2: Illustrations to the proof of Corollary \ref{['cor:ineq-pl-pat-dsp']}.
• Figure 3: Illustrations to the proof of Lemma \ref{['lm:pat_fat']}. (a) a fat $K_3$-minor. (b) a fat $K_{1,3}$-minor.

Theorems & Definitions (46)

• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• corollary thmcountercorollary
• proposition thmcounterproposition: DrKoLe2017
• theorem thmcountertheorem
• lemma thmcounterlemma