The matching preclusion number of a graph G, mp(G), is the minimum number of edges whose deletion leaves a resulting graph that has neither perfect matchings nor almost perfect matchings. Besides its theoretical linkage with conditional connectivity and extremal graph theory, the matching preclusion number is a measure of robustness in interconnection networks. In this paper we develop general ...

The purpose of the independent set interdiction problem in the weighted graph G is to determine a set of vertices R∗ such that the weight of the maximum independent set in G−R∗ is minimized. We define an approximate solution for this problem. Then, an upper bound for the relative error of this problem is obtained. We show that the limit of the relative error of the independent set interdiction ...

In this paper, we present a new method to produce explicit formulas for the number of rooted spanning forests $f(n)$ infinite family graphs $H_n=H_n(G_1,G_2,\ldots,G_m)$ obtained as circulant foliation over graph $H$ on $m$ vertices with fibers $G_1,G_2,\ldots,G_m.$ Each fiber $G_i=C_n(s_{i,1},s_{i,2},\ldots,s_{i,k_i})$ is $n$ jumps $s_{i,1},s_{i,2},\ldots,s_{i,k_i}.$ This includes generalized ...

A/Mmcr-It is proved that the Shannon zero-error capacity of the pentagon is e. The method is then generalized to obtain upper bounds on the capacity of au arbitrary graph. A well-characterized, and in a sense easily computable, function is introduced which bounds the capacity from above and equals the capacity in a large number of cases. Several results are obtained on the capacity of special g...

In two recent papers [Hafner, Elec. J. Combin. 11 (R77) (2004), 1–33; Ilić, Pace and Magliveras, J. Combin. Math. Combin. Comput. 80 (2012), 267–275] it was shown that the Higman-Sims graph Γ can be decomposed into a disjoint union of five double Petersen graphs. In the second of these papers, it was further shown that all such decompositions fall into a single orbit under the action of sporadi...

Let G = (V, E) be a graph. A function f : V → {−1,+1} defined on the vertices of G is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. The signed total domination number of G, γ t (G), is the minimum weight of a signed total dominating function of G. In this paper, we study the signed total domination number of generalized Petersen...

A graph G is said to be distance-balanced if for any edge uv of G, the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. Let GP (n, k) be a generalized Petersen graph. Jerebic, Klavžar, and Rall [Distance-balanced graphs, Ann. Comb. 12 (2008) 71–79] conjectured that: For any integer k > 2, there exists a positive integer n0 such that the GP (n, k...

The {\em topological symmetry group} of an embedding $\Gamma$ abstract graph $\gamma$ in $S^3$ is the group automorphisms which can be realized by homeomorphisms pair $(S^3, \Gamma)$. These groups are motivated questions about symmetries molecules space. Petersen family graphs important for many problems low dimensional topology, so it desirable to understand possible their embeddings In this p...