Characterizing forbidden pairs for spanning $\Theta$-subgraphs of 2-connected graphs

Abstract

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{$\Theta$-graph} is the graph consisting of three internally disjoint paths with the same pair of end-vertices. If the $\Theta$-subgraph $T$ contains all vertices of $G$, then we call $T$ a \emph{spanning $\Theta$-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 $\Theta$-subgraph. In order to obtain this result, we also characterize all minimal 2-connected non-cycle claw-free graphs without spanning $\Theta$-subgraphs.