The average size of a connected vertex set of a k-connected graph

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(...

connected cototal domination number of a graph

a dominating set $d subseteq v$ of a graph $g = (v,e)$ is said to be a connected cototal dominating set if $langle d rangle$ is connected and $langle v-d rangle neq phi$, contains no isolated vertices. a connected cototal dominating set is said to be minimal if no proper subset of $d$ is connected cototal dominating set. the connected cototal domination number $gamma_{ccl}(g)$ of $g$ is the min...

Optimal Augmentation of a 2-Vertex-Connected Multigraph to a k-Edge-Connected and 3-Vertex-Connected Multigraph

The connected vertex detour number of a graph

For a connected graph G of order p ≥ 2 and a vertex x of G, a set S ⊆ V(G) is an x-detour set of G if each vertex v ∈ V(G) lies on an x − y detour for some element y in S. The minimum cardinality of an xdetour set of G is defined as the x-detour number of G, denoted by dx(G). An x-detour set of cardinality dx(G) is called a dx-set of G. A connected x-detour set of G is an x-detour set S such th...

The Connected Vertex Monophonic Number of a Graph

For a connected graph G of order p ≥ 2 and a vertex x of G, a set S ⊆ V (G) is an x-monophonic set of G if each vertex v ∈ V (G) lies on an x− y monophonic path for some element y in S. The minimum cardinality of an x-monophonic set of G is defined as the x-monophonic number of G, denoted by mx(G). A connected x-monophonic set of G is an x-monophonic set S such that the subgraphG[S] induced by ...