A graph G is called preperfect if each induced subgraph G 0 G of order at least 2 has two vertices x; y such that either all maximum cliques of G 0 containing x contain y, or all maximum independent sets of G 0 containing y contain x, too. Giving a partial answer to a problem of Hammer and Maaray Combinatorica 13 (1993), 199{208], we describe new classes of minimally non-preperfect graphs, and ...

Krishnamoorthy, Thulasiraman and Swamy [Minimum order graphs with specified diameter, connectivity and regularity, Networks 19 (1989) 25–46] showed that a δ-regular graph with diameter D at most 3 has (vertex-)connectivity κ at least 2, and if D ≤ 2 then the connectivity is at least κ ≥ min{δ, 3}. Likewise, Soneoka, Nakada, Imase and Peyrat [Sufficient conditions for maximally connected graphs,...

A connected graph is said to be super edge-connected if every minimum edge-cut isolates a vertex. The restricted edge-connectivity λ′ of a connected graph is the minimum number of edges whose deletion results in a disconnected graph such that each connected component has at least two vertices. A graph G is called λ′-optimal if λ′(G) = min{dG(u)+dG(v)−2 : uv is an edge in G}. This paper proves t...

A graph is k-connected if it has k internally-disjoint paths between every pair of nodes. A subset S of nodes in a graph G is a kconnected set if the subgraph G[S] induced by S is k-connected; S is an m-dominating set if every v ∈ V \ S has at least m neighbors in S. If S is both k-connected and m-dominating then S is a k-connected m-dominating set, or (k,m)-cds for short. In the k-Connected mD...

The concept of the bipartite divisor graph for integer subsets has been considered in [M. A. Iranmanesh and C. E. Praeger, Bipartite divisor graphs for integer subsets, Graphs Combin., 26 (2010) 95–105.]. In this paper, we will consider this graph for the set of character degrees of a finite group G and obtain some properties of this graph. We show that if G is a solvable group, then the number...

hansen et‎. ‎al.‎, ‎using the autographix software ‎package‎, ‎conjectured that the szeged index $sz(g)$ and the‎ ‎wiener index $w(g)$ of a connected bipartite graph $g$ with $n geq ‎4$ vertices and $m geq n$ edges‎, ‎obeys the relation‎ ‎$sz(g)-w(g) geq 4n-8$‎. ‎moreover‎, ‎this bound would be the best possible‎. ‎this paper offers a proof to this conjecture‎.

We give a simple proof of Tutte’s theorem stating that the cycle space of a 3–connected graph is generated by the set of non-separating circuits of the graph.

In the recent years, non-connected piled raft foundation has been considered as an economical and practical deep foundation in the situation that high shear and concentrated loads may occur at the connection of the raft and pile head. This paper was presented an experimental study of a square foundation on the effects of parameters such as S/D, L/D and etc. in two cases of connected or non-conn...

In this article we will present a graph partitioning algorithm which partitions a graph into two different types of components: the well-known ‘strongly connected components’ as well as another type of components we call ‘connected acyclic component’. We will give an algorithm based on Tarjan’s algorithm for finding strongly connected components used to find such a partitioning. We will also sh...