Magdalena Lemańska

On the doubly connected domination number of a graph

AbstractFor a given connected graph G = (V, E), a set

On the partition dimension of trees

D \subseteq V(G)

Nordhaus-Gaddum results for weakly convex domination number of a graph

is a doubly connected dominating set if it is dominating and both 〈D〉 and 〈V (G)-D〉 are connected. The cardinality of the minimum doubly connected dominating set in G is the doubly connected domination number. We investigate several properties of doubly connected dominating sets and give some bounds on the doubly connected domination number.

Domination Numbers in Graphs with Removed Edge or Set of Edges

The bondage number b(G) of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph with domination number greater than the domination number of G. Here we study the bondage number of some grid-like graphs. In this sense, we obtain some bounds or exact values of the bondage number of some strong product and direct product of two paths.

Some Variations of Perfect Graphs

Given an ordered partition ? = { P 1 , P 2 , ? , P t } of the vertex set V of a connected graph G = ( V , E ) , the partition representation of a vertex v ? V with respect to the partition ? is the vector r ( v | ? ) = ( d ( v , P 1 ) , d ( v , P 2 ) , ? , d ( v , P t ) ) , where d ( v , P i ) represents the distance between the vertex v and the set P i . A partition ? of V is a resolving partition of G if different vertices of G have different partition representations, i.e., for every pair of vertices u , v ? V , r ( u | ? ) ? r ( v | ? ) . The partition dimension of G is the minimum number of sets in any resolving partition of G . In this paper we obtain several tight bounds on the partition dimension of trees.

Total domination multisubdivision number of a graph

For a connected graph G = (V,E), a set D ⊆ V (G) is a dominating set of G if every vertex in V (G)−D has at least one neighbour in D. The distance dG(u, v) between two vertices u and v is the length of a shortest (u− v) path in G. An (u− v) path of length dG(u, v) is called an (u− v)-geodesic. A set X ⊆ V (G) is convex in G if vertices from all (a − b)-geodesics belong to X for any two vertices a, b ∈ X. A set X is a convex dominating set if it is convex and dominating. The convex domination number γcon(G) of a graph G is the minimum cardinality of a convex dominating set in G. Graphs with the convex domination number close to their order are studied. The convex domination number of a Cartesian product of graphs is also considered.

Total Domination Versus Domination in Cubic Graphs

Nordhaus-Gaddum results for weakly convex domination number of a graph G are studied.

Relations between edge removing and edge subdivision concerning domination number of a graph

It is known that the removal of an edge from a graph G cannot decrease a domination number γ(G) and can increase it by at most one. Thus we can write that γ(G) ≤ γ(G − e) ≤ γ(G) + 1 when an arbitrary edge e is removed. Here we present similar inequalities for the weakly connected domination number γw and the connected domination number γc, i.e., we show that γw(G) ≤ γw(G− e) ≤ γw(G) + 1 and γc(G) ≤ γc(G − e) ≤ γc(G) + 2 if G and G − e are connected. Additionally we show that γw(G) ≤ γw(G−Ep) ≤ γw(G) + p− 1 and γc(G) ≤ γc(G− Ep) ≤ γc(G) + 2p− 2 if G and G− Ep are connected and Ep = E(Hp) where Hp of order p is a connected subgraph of G.