A special class of cubic graphs are the cycle permutation graphs. A cycle permutation graph Pn(α) is defined by taking two vertex-disjoint cycles on n vertices and adding a matching between the vertices of the two cycles.In this paper we determine a good upper bound for tenacity of cycle permutation graphs.

For a k-connected graph or matroid M , where k is a fixed positive integer, we say that a subset X of E(M) is k-removable provided M\X is k-connected. In this paper, we obtain a sharp condition on the size of a 3connected binary matroid to have a 3-removable circuit.

Let {Gp1, Gp2, . . .} be an infinite sequence of graphs with Gpn having pn vertices. This sequence is called Kp-removable if Gp1 ∼= Kp, and Gpn − S ∼= Gp(n−1) for every n ≥ 2 and every vertex subset S of Gpn that induces a Kp. Each graph in such a sequence has a high degree of symmetry: every way of removing the vertices of any fixed number of disjoint Kp’s yields the same subgraph. Here we con...

An edge e of a k-connected graph G is said to be a removable edge if G a e is still k-connected, where Ga e denotes the graph obtained from G by deleting e to get G− e and, for any end vertex of e with degree k− 1 in G− e, say x, deleting x and then adding edges between any pair of non-adjacent vertices in NG−e(x). Xu and Guo [28] proved that a 5-connected graph G has no removable edge if and o...

Kriesell proved that every almost critical graph of connectivity 2 nonisomorphic to a cycle has at least 2 removable ears of length greater than 2. We improve this lower bound on the number of removable ears. A necessary condition for critically 2-connected graphs in terms of a forbidden minor is obtained. Further, we investigate properties of a special class of critically 2-connected series-pa...

A graph G = (V,E) is called l-mixed p-connected if G−S−L is connected for all pairs S, L with S ⊆ V , L ⊆ E, and l|S|+ |L| < p. This notion is a common generalisation of m-vertex-connectivity (l = 1, p = m) and m-edge-connectivity (l ≥ m, p = m). If p = kl then we obtain (k, l)-connectivity, introduced earlier by Kaneko and Ota, as a special case. We show that by using maximum adjacency orderin...

A cycle C is a vertex-dominating cycle if every vertex is adjacent to some vertex of C. Bondy and Fan [4] showed that if G is a 2-connected graph with δ(G) ≥ 13 (|V (G)| − 4), then G has a vertex-dominating cycle. In this paper, we prove that if G is a 2-connected bipartite graph with partite sets V1 and V2 such that δ(G) ≥ 1 3 (max{|V1|, |V2|}+ 1), then G has a vertex-dominating cycle.

In this paper, we continue the study of the domination game in graphs introduced by Bre{v{s}}ar, Klav{v{z}}ar, and Rall. We study the paired-domination version of the domination game which adds a matching dimension to the game. This game is played on a graph $G$ by two players, named Dominator and Pairer. They alternately take turns choosing vertices of $G$ such that each vertex chosen by Domin...

We present a matching and LP based heuristic algorithm that decides graph non-Hamiltonicity. Each of the n! Hamilton cycles in a complete directed graph on n + 1 vertices corresponds with each of the n! n-permutation matrices P, such that pu,i = 1 if and only if the ith arc in a cycle enters vertex u, starting and ending at vertex n + 1. A graph instance (G) is initially coded as exclusion set ...