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

2025-10-20

AI Summary

This paper investigates the longest path transversal number (lpt) of a graph—the minimum number of vertices required to intersect all its longest paths. We develop a unified analytical framework integrating domination theory, graph decomposition, and intersection graph models, applicable to hereditary graph classes—including $P_t$-free graphs, (bull, chair)-free graphs, chordal graphs—and $H$-graphs. Our key contribution is establishing, for the first time, **constant upper bounds** on lpt for multiple structurally constrained graph families: $mathrm{lpt} leq 3$ for connected $P_5$-free graphs; $mathrm{lpt} leq 4$ for connected $P_6$-free graphs; $mathrm{lpt} leq 5$ for (bull, chair)-free graphs; and, most generally, $mathrm{lpt} leq k(H)$ for any connected $H$-graph, where $k(H)$ depends only on the fixed graph $H$. These results resolve long-standing open questions regarding the boundedness of lpt across several fundamental graph classes.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The extit{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 extit{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 extit{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 $( extit{bull}, extit{chair})$-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 extit{$H$-intersection graphs}. For a given graph $H$, a graph $G$ is called an extit{$H$-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 $lpt(G) leq k$.

Problem

• Determine upper bounds for longest path transversal number in hereditary graph classes
• Establish constant bounds for longest path transversal in specific H-free graphs
• Prove bounded longest path transversal numbers exist for all H-graph families

Innovation

• Refining longest path transversals using domination properties
• Establishing constant bounds for P_t-free and bull-chair-free graphs
• Proving bounded transversal numbers for H-intersection graphs