Path Eccentricity and Forbidden Induced Subgraphs
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2025
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| _version_ | 1866909745153048576 |
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| author | Cichacz, Sylwia Hilaire, Claire Masařík, Tomáš Masaříková, Jana Milanič, Martin |
| author_facet | Cichacz, Sylwia Hilaire, Claire Masařík, Tomáš Masaříková, Jana Milanič, Martin |
| contents | 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 $\{\text{claw}, \text{net}\}$-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 Bacsó 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 $P_{2k+1} + P_{k-1}$. For such cases, we also provide a robust polynomial-time algorithm that finds a path witnessing the upper bound on the path eccentricity. Our main result also answers an open question of Bastide, Hilaire, and Robinson [Discrete Math., 2025]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_15747 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Path Eccentricity and Forbidden Induced Subgraphs Cichacz, Sylwia Hilaire, Claire Masařík, Tomáš Masaříková, Jana Milanič, Martin Combinatorics Discrete Mathematics 05C12, 05C38 G.2.2 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 $\{\text{claw}, \text{net}\}$-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 Bacsó 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 $P_{2k+1} + P_{k-1}$. For such cases, we also provide a robust polynomial-time algorithm that finds a path witnessing the upper bound on the path eccentricity. Our main result also answers an open question of Bastide, Hilaire, and Robinson [Discrete Math., 2025]. |
| title | Path Eccentricity and Forbidden Induced Subgraphs |
| topic | Combinatorics Discrete Mathematics 05C12, 05C38 G.2.2 |
| url | https://arxiv.org/abs/2503.15747 |