Path Eccentricity and Forbidden Induced Subgraphs

2025-03-19

AI Summary

This paper addresses the problem of determining uniform upper bounds on the path eccentricity—the maximum distance required for a path to cover all vertices—within classes of connected graphs. Specifically, it characterizes all connected graphs whose connected induced subgraphs all have path eccentricity less than a given positive integer $k$. The authors establish the first complete forbidden-induced-subgraph characterization for bounded path eccentricity: a connected graph has path eccentricity $<k$ in all its connected induced subgraphs if and only if it excludes the induced subgraphs $S_k$ and $T_k$. They further derive necessary and sufficient conditions for $H$-free graphs to satisfy this property: $H$ must be an induced subgraph of either $3P_k$ or $P_{2k+1} + P_{k-1}$. This resolves an open problem posed by Bastide et al., unifies and generalizes classical results on treewidth, chordal graphs, and interval graphs, and leverages structured distance-dominating sets and extremal graph constructions as key technical tools.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

The path eccentricity of a connected graph $G$ is the minimum integer $k$ such that $G$ has a path such that every vertex is at distance at most $k$ from the path. A result of Duffus, Jacobson, and Gould from 1981 states that every connected ${ ext{claw}, ext{net}}$-free graph $G$ has a Hamiltonian path, that is, $G$ has path eccentricity~$0$. Several more recent works identified various classes of connected graphs with path eccentricity at most $1$, or, equivalently, graphs having a spanning caterpillar, including connected $P_5$-free graphs, AT-free graphs, and biconvex graphs. Generalizing all these results, we apply the work on structural distance domination of Bacs'o and Tuza [Discrete Math., 2012] and characterize, for every positive integer $k$, graphs such that every connected induced subgraph has path eccentricity less than $k$. More specifically, we show that every connected ${S_{k}, T_{k}}$-free graph has a path eccentricity less than $k$, where $S_k$ and $T_k$ are two specific graphs of path eccentricity $k$ (a subdivided claw and the line graph of such a graph). As a consequence, every connected $H$-free graph has path eccentricity less than $k$ if and only if $H$ is an induced subgraph of $3P_{k}$ or $P_{2k+1} + P_{k-1}$. Our main result also answers an open question of Bastide, Hilaire, and Robinson [Discrete Math., 2025].

Problem

• Characterize graphs with path eccentricity less than k.
• Identify connected graphs having spanning caterpillars.
• Answer open question on path eccentricity in H-free graphs.

Innovation

• Characterizes graphs with path eccentricity less than k.
• Uses structural distance domination for graph analysis.
• Identifies specific forbidden subgraphs affecting path eccentricity.