Matching Preclusion of the Generalized Petersen Graph

The matching preclusion number of a graph with an even number of vertices is the minimum number of edges whose deletion results in a graph with no perfect matchings. In this paper we determine the matching preclusion number for the generalized Petersen graph $P(n,k)$ and classify the optimal sets

A lower bound for the 3-pendant tree-connectivity of lexicographic product graphs

summary:For a connected graph $G=(V,E)$ and a set $S \subseteq V(G)$ with at least two vertices, an $S$-Steiner tree is a subgraph $T = (V',E')$ of $G$ that is a tree with $S \subseteq V'$. If the degree of each vertex of $S$ in $T$ is equal to 1, then $T$ is called a pendant $S$-Steiner tree. Two $S$-Steiner trees are {\it internally disjoint} if they share no vertices other than $S$ and have no edges in common. For $S\subseteq V(G)$ and $|S|\geq 2$, the pendant tree-connectivity $\tau _G(S)$ is the maximum number of internally disjoint pendant $S$-Steiner trees in $G$, and for $k \geq 2$, the $k$-pendant tree-connectivity $\tau _k(G)$ is the minimum value of $\tau _G(S)$ over all sets $S$ of $k$ vertices. We derive a lower bound for $\tau _3(G\circ H)$, where $G$ and $H$ are connected graphs and $\circ$ denotes the lexicographic product

Fractional matching preclusion for generalized augmented cubes

The \emph{matching preclusion number} of a graph is the minimum number ofedges whose deletion results in a graph that has neither perfect matchings noralmost perfect matchings. As a generalization, Liu and Liu recently introducedthe concept of fractional matching preclusion number. The \emph{fractionalmatching preclusion number} of $G$ is the minimum number of edges whosedeletion leaves the resulting graph without a fractional perfect matching. The\emph{fractional strong matching preclusion number} of $G$ is the minimumnumber of vertices and edges whose deletion leaves the resulting graph withouta fractional perfect matching. In this paper, we obtain the fractional matchingpreclusion number and the fractional strong matching preclusion number forgeneralized augmented cubes. In addition, all the optimal fractional strongmatching preclusion sets of these graphs are categorized.Comment: 21 pages; 1 figure