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

TL;DR

This work characterizes which pairs of connected forbidden subgraphs force the existence of a spanning $\\Theta$-subgraph in every 2-connected graph, focusing on nontrivial cases excluding $P_3$. It introduces an unfoldment framework that translates claw-free graph structure into loopless and semi-loopless multigraphs, linking the existence of spanning $\\Theta$-subgraphs to Euler tours in the underlying multigraphs. The authors prove a precise classification of minimal 2-connected non-cycle claw-free graphs without spanning $\\Theta$-subgraphs, showing they belong to seven families $\\\mathcal{H}_i$, and use this to derive a complete list of forbidden pairs: up to symmetry, either $(R,S)=(K_{1,4},P_4)$ or $(K_{1,3},S)$ with $S$ an induced subgraph of $B_{1,5}, B_{2,4}, N_{1,1,4}$, or $N_{1,2,3}$. These results extend the theory of forbidden induced subgraphs for spanning substructures and provide a structural framework for analyzing Theta-spanning properties in 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{$\varTheta$-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.

Figures (10)

• Figure 1: Graphs $B_{1,5}$, $B_{2,4}$, $N_{1,1,4}$ and $N_{1,2,3}$.
• Figure 2: Classes of links $\mathcal{L}_1(x,y)$ and $\mathcal{L}_2(x,y)$.
• Figure 3: A chain from $\{x_1,y_1\}$ to $\{x_2,y_2\}$.
• Figure 4: Classes of graphs $\mathcal{H}_1, \mathcal{H}_2, \mathcal{H}_3, \mathcal{H}_4$.
• Figure 5: Classes $\mathcal{H}_5$, $\mathcal{H}_6$ and $\mathcal{H}_7$.

Theorems & Definitions (53)

• Theorem 1: Faudree and Gould 5
• Theorem 2
• Theorem 3: Brousek B1
• Theorem 4
• Theorem 5: Oberly, Simic and Sumner 6
• Theorem 6: Egawa 4
• Theorem 7: Chvátal and Erdős 3
• Lemma 1
• proof
• Lemma 2