Solid bricks that every $b$-invariant edge is solitary

TL;DR

This paper characterizes solid bricks in graph theory, proving that if every $b$-invariant edge is solitary in such a graph, then it must be a wheel graph, providing a complete classification.

Contribution

It establishes a precise characterization of solid bricks with solitary $b$-invariant edges as wheel graphs, resolving a problem posed by Lucchesi and Murty.

Findings

Solid bricks with all $b$-invariant edges solitary are exactly wheel graphs.

The result excludes $K_4$, $ar{C}_6$, and Petersen graph from this characterization.

Provides a complete classification of such graphs in the context of graph theory.

Abstract

A graph $G$ is a brick if it is 3-connected and $G-\{u,v\}$ has a perfect matching for any two distinct vertices $u$ and $v$ of $G$. A brick $G$ is solid if for any two vertex disjoint odd cycles $C_1$ and $C_2$ of $G$, $G-(V(C_1)\cup V(C_2))$ has no perfect matching. Lucchesi and Murty proposed a problem concerning the characterization of bricks, distinct from $K_4$, $\overline{C_6}$ and the Petersen graph, in which every $b$-invariant edge is solitary. In this paper, we show that for a solid brick $G$ of order $n$ that is distinct from $K_4$, every $b$-invariant edge of $G$ is solitary if and only if $G$ is a wheel $W_n$.