In this paper, we study both concepts of geodetic dominatingand edge geodetic dominating sets and derive some tight upper bounds onthe edge geodetic and the edge geodetic domination numbers. We also obtainattainable upper bounds on the maximum number of elements in a partitionof a vertex set of a connected graph into geodetic sets, edge geodetic sets,geodetic domin...

A set C of vertices in a graph G = (V,E) is total dominating in G if all vertices of V are adjacent to a vertex of C. Furthermore, if a total dominating set C in G has the additional property that for any distinct vertices u, v ∈ V \ C the subsets formed by the vertices of C respectively adjacent to u and v are different, then we say that C is a locating-total dominating set in G. Previously, l...

Let G be a graph with vertex set V (G), and let f : V (G) −→ {−1, 1} be a two-valued function. If k ≥ 1 is an integer and ∑ x∈N(v) f(x) ≥ k for each v ∈ V (G), where N(v) is the neighborhood of v, then f is a signed total k-dominating function on G. A set {f1, f2, . . . , fd} of distinct signed total k-dominating functions on G with the property that ∑d i=1 fi(x) ≤ k for each x ∈ V (G), is call...

This paper is devoted to the online dominating set problem and its variants on trees, bipartite, bounded-degree, planar, and general graphs, distinguishing between connected and not necessarily connected graphs. We believe this paper represents the first systematic study of the effect of two limitations of online algorithms: making irrevocable decisions while not knowing the future, and being i...

In this paper we study cooperative cost games arising from domination problems on graphs. We introduce three games to model the cost allocation problem and we derive a necessary and sufficient condition for the balancedness of all three games. Furthermore we study concavity of these games.

We study the approximability of the weighted edge-dominating set problem. Although even the unweighted case is NP-Complete, in this case a solution of size at most twice the minimum can be efficiently computed due to its close relationship with minimum maximal matching; however, in the weighted case such a nice relationship is not known to exist. In this paper, after showing that weighted edge ...

A set C of vertices in a graph G = (V,E) is total dominating in G if all vertices of V are adjacent to a vertex of C. Furthermore, if a total dominating set C in G has the additional property that for any distinct vertices u, v ∈ V \ C the subsets formed by the vertices of C respectively adjacent to u and v are different, then we say that C is a locating-total dominating set in G. Previously, l...

A set C of vertices in a graph G = (V,E) is total dominating in G if all vertices of V are adjacent to a vertex of C. Furthermore, if a total dominating set C in G has the additional property that for any distinct vertices u, v ∈ V \ C the subsets formed by the vertices of C respectively adjacent to u and v are different, then we say that C is a locating-total dominating set in G. Previously, l...

Let $G$ be a simple graph with vertex set $V$. A double Roman dominating function (DRDF) on $G$ is a function $f:Vrightarrow{0,1,2,3}$ satisfying that if $f(v)=0$, then the vertex $v$ must be adjacent to at least two vertices assigned $2$ or one vertex assigned $3$ under $f$, whereas if $f(v)=1$, then the vertex $v$ must be adjacent to at least one vertex assigned $2$ or $3$. The weight of a DR...