Isometric path partition: a new upper bound and a characterization of some extremal graphs
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866918034841534464 |
|---|---|
| author | Penev, Irena Sandeep, R. B. Supraja, D. K. Taruni, S. |
| author_facet | Penev, Irena Sandeep, R. B. Supraja, D. K. Taruni, S. |
| contents | An $\textit{isometric path}$ is a shortest path between two vertices. An $\textit{isometric path partition}$ (IPP) of a graph $G$ is a set $I$ of vertex-disjoint isometric paths in $G$ that partition the vertices of $G$. The \textit{isometric path partition number} of $G$, denoted by $\text{ipp}(G)$, is the minimum cardinality of an IPP of $G$. In this article, we prove that every graph $G$ satisfies $\text{ipp}(G) \leq |V(G)| - ν(G)$, where $ν(G)$ is matching number of $G$. We further prove that a connected graph $G$ is extremal with respect to this upper bound, i.e.\ satisfies $\text{ipp}(G) = |V(G)| - ν(G)$, if and only if either (i) all blocks of $G$ are odd complete graphs, or (ii) all blocks of $G$ except one are odd complete graphs, and the unique block $B$ of $G$ that is not an odd complete graph is even and satisfy $\text{ipp}(B) = |V(B)| - ν(B)$. As corollaries of this result, we obtain a full structural characterization of all connected odd graphs that are extremal with respect to our upper bound, as well as of all extremal block graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19913 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Isometric path partition: a new upper bound and a characterization of some extremal graphs Penev, Irena Sandeep, R. B. Supraja, D. K. Taruni, S. Combinatorics An $\textit{isometric path}$ is a shortest path between two vertices. An $\textit{isometric path partition}$ (IPP) of a graph $G$ is a set $I$ of vertex-disjoint isometric paths in $G$ that partition the vertices of $G$. The \textit{isometric path partition number} of $G$, denoted by $\text{ipp}(G)$, is the minimum cardinality of an IPP of $G$. In this article, we prove that every graph $G$ satisfies $\text{ipp}(G) \leq |V(G)| - ν(G)$, where $ν(G)$ is matching number of $G$. We further prove that a connected graph $G$ is extremal with respect to this upper bound, i.e.\ satisfies $\text{ipp}(G) = |V(G)| - ν(G)$, if and only if either (i) all blocks of $G$ are odd complete graphs, or (ii) all blocks of $G$ except one are odd complete graphs, and the unique block $B$ of $G$ that is not an odd complete graph is even and satisfy $\text{ipp}(B) = |V(B)| - ν(B)$. As corollaries of this result, we obtain a full structural characterization of all connected odd graphs that are extremal with respect to our upper bound, as well as of all extremal block graphs. |
| title | Isometric path partition: a new upper bound and a characterization of some extremal graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2505.19913 |