The Connected Detour Numbers of Special Classes of Connected Graphs

The connected forcing connected vertex detour number of a graph

For any vertex x in a connected graph G of order p ≥ 2, a set S of vertices of V is an x-detour set of G if each vertex v in G lies on an x-y detour for some element y in S. A connected x-detour set of G is an x-detour set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected x-detour set of G is the connected x-detour number of G and is denoted by cdx(...

Generating particular classes of connected graphs

We present an algorithm to generate all connected graphs in a recursive and efficient manner. This algorithm is restricted subsequently to 1-particle irreducible graphs and to connected graphs without selfloops. The recursions proceed by loop order and vertex number. The main result of [1] is a recursion formula to generate all tree graphs. The underlying structure is a Hopf algebraic represent...

On Distance Connected Domination Numbers of Graphs

Let k be a positive integer and G = (V,E) be a connected graph of order n. A set D ⊆ V is called a k-dominating set of G if each x ∈ V (G) − D is within distance k from some vertex of D. A connected k-dominating set is a k-dominating set that induces a connected subgraph of G. The connected k-domination number of G, denoted by γ k(G), is the minimum cardinality of a connected k-dominating set. ...

The maximal total irregularity of some connected graphs

The total irregularity of a graph G is defined as 〖irr〗_t (G)=1/2 ∑_(u,v∈V(G))▒〖|d_u-d_v |〗, where d_u denotes the degree of a vertex u∈V(G). In this paper by using the Gini index, we obtain the ordering of the total irregularity index for some classes of connected graphs, with the same number of vertices.

Spanning 2-Connected Subgraphs in Alphabet Graphs, Special Classes of Grid Graphs