Characterizing forbidden pairs for spanning $\varTheta$-subgraphs of 2-connected graphs
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915197831086080 |
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| author | Li, Binlong Sang, Ziqing Wang, Shipeng |
| author_facet | Li, Binlong Sang, Ziqing Wang, Shipeng |
| contents | Let $\mathcal{F}$ be a set of connected graphs, and let $G$ be a graph. We say that $G$ is \emph{$\mathcal{F}$-free} if it does not contain $F$ as an induced subgraph for all $F\in\mathcal{F}$, and we call $\mathcal{F}$ a forbidden pair if $|\mathcal{F}|=2$. A \emph{$\varTheta$-graph} is the graph consisting of three internally disjoint paths with the same pair of end-vertices. If the $\varTheta$-subgraph $T$ contains all vertices of $G$, then we call $T$ a \emph{spanning $\varTheta$-subgraph} of $G$. In this paper, we characterize all pairs of connected graphs $R,S$ such that every 2-connected $\{R,S\}$-free graph has a spanning $\varTheta$-subgraph. In order to obtain this result, we also characterize all minimal 2-connected non-cycle claw-free graphs without spanning $\varTheta$-subgraphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_11176 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterizing forbidden pairs for spanning $\varTheta$-subgraphs of 2-connected graphs Li, Binlong Sang, Ziqing Wang, Shipeng Combinatorics Let $\mathcal{F}$ be a set of connected graphs, and let $G$ be a graph. We say that $G$ is \emph{$\mathcal{F}$-free} if it does not contain $F$ as an induced subgraph for all $F\in\mathcal{F}$, and we call $\mathcal{F}$ a forbidden pair if $|\mathcal{F}|=2$. A \emph{$\varTheta$-graph} is the graph consisting of three internally disjoint paths with the same pair of end-vertices. If the $\varTheta$-subgraph $T$ contains all vertices of $G$, then we call $T$ a \emph{spanning $\varTheta$-subgraph} of $G$. In this paper, we characterize all pairs of connected graphs $R,S$ such that every 2-connected $\{R,S\}$-free graph has a spanning $\varTheta$-subgraph. In order to obtain this result, we also characterize all minimal 2-connected non-cycle claw-free graphs without spanning $\varTheta$-subgraphs. |
| title | Characterizing forbidden pairs for spanning $\varTheta$-subgraphs of 2-connected graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.11176 |