Hitting all longest paths in $H$-free graphs and $H$-graphs

TL;DR

The paper investigates constant upper bounds for the longest path transversal number $\mathsf{lpt}(G)$ across graph classes. It develops a structural refinement tool that uses domination to adjust transversals and yields constant bounds for several hereditary classes, notably $lpt(G)\le t-2$ for $P_t$-free graphs with $t\in\{5,6\}$, $lpt(G)\le 5$ for $(bull,chair)$-free graphs, and $lpt(G)\le t-1$ for chordal graphs forbidding $K_t\boxminus\overline{K_t}$; it also establishes a universal bound for connected $H$-graphs, $\mathsf{lpt}(G)\le 4(\mathsf{tw}(H)+1)|E(H)|$, via an $H$-profile and an intermediate graph $\widehat{H}$. The approach links domination, monitors, and Helly-type arguments with $H$-representations to obtain these results. Collectively, the findings extend constant upper bounds to broad families including interval, circular-arc, and chordal graphs through the $H$-graph framework, and provide a concrete dependence of the bound on the structure of $H$.

Abstract

The \textit{longest path transversal number} of a connected graph $G$, denoted by $lpt(G)$, is the minimum size of a set of vertices of $G$ that intersects all longest paths in $G$. We present constant upper bounds for the longest path transversal number of \textit{hereditary classes of graphs}, that is, classes of graphs closed under taking induced subgraphs. Our first main result is a structural theorem that allows us to \textit{refine} a given longest path transversal in a graph using domination properties. This has several consequences: First, it implies that for every $t \in \{5,6\}$, every connected $P_t$-free graph $G$ satisfies $lpt(G) \leq t-2$. Second, it shows that every $(\textit{bull}, \textit{chair})$-free graph $G$ satisfies $lpt(G) \leq 5$. Third, it implies that for every $t \in \mathbb{N}$, every connected chordal graph $G$ with no induced subgraph isomorphic to $K_t \mat \overline{K_t}$ satisfies $lpt(G) \leq t-1$, where $K_t \mat \overline{K_t}$ is the graph obtained from a $t$-clique and an independent set of size $t$ by adding a perfect matching between them. Our second main result provides an upper bound for the longest path transversal number in \textit{$H$-intersection graphs}. For a given graph $H$, a graph $G$ is called an \textit{$H$-graph} if there exists a subdivision $H'$ of $H$ such that $G$ is the intersection graph of a family of vertex subsets of $H'$ that each induce connected subgraphs. The concept of $H$-graphs, introduced by Biró, Hujter, and Tuza, naturally captures interval graphs, circular-arc graphs, and chordal graphs, among others. Our result shows that for every connected graph $H$ with at least two vertices, there exists an integer $k = k(H)$ such that every connected $H$-graph $G$ satisfies $lpt(G) \leq k$.

Figures (5)

• Figure 1: The Walther-Zamfirescue graph.
• Figure 2: From left to right: a chair, a bull, and a $K_4 \boxminus \overline{K_4}$.
• Figure 3: From left to right: A graph $G$, a graph $H$, and an $H$-representation of $G$. Vertices in white color are correspond to the subdivided edges of $H$.
• Figure 4: Outcome of \ref{['lem:monitor']} for $t=6$. For $i\in [3]$, $C_i$'s are connected components in $G\setminus N_{G}[X]$.
• Figure 5: An $(R,D,T)$ decomposition. $C_i$'s are components in $G - (R\cup D)$. Red lines represent complete adjacency between $R$ and $D$.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 3.1
• proof
• Lemma 3.2: Chudnovsky, King, Pilipczuk, Rzążewski, and Spirkl; see Lemma 5 in chudnovsky2021
• Theorem 3.3
• proof