A {em Roman dominating function} on a graph $G$ is a function$f:V(G)rightarrow {0,1,2}$ satisfying the condition that everyvertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex$v$ for which $f(v) =2$. {color{blue}A {em restrained Roman dominating}function} $f$ is a {color{blue} Roman dominating function if the vertices with label 0 inducea subgraph with no isolated vertex.} The wei...

We consider the problem of finding an orientation with minimum diameter of a connected bridgeless graph. Fomin et. al. [7] discovered a relation between the minimum oriented diameter an the size of a minimal dominating set. We improve their upper bound.

A subset S of V is called a dominating set in G if every vertex in V-S is adjacent to at least one vertex in S. A Dominating set is said to be Fuzzy Double Dominating set if every vertex in V-S is adjacent to at least two vertices in S. The minimum cardinality taken over all, the minimal double dominating set is called Fuzzy Double Domination Number and is denoted by γ fdd (G). The minimum numb...

Let 7(G) denote the minimum cardinality of a dominating set of a graph G = (V,E). A longstanding upper bound for 7(G) is attributed to Berge: For any graph G with n vertices and maximum degree A(G), 7(G) <~ n A(G). We eharacterise connected bipartite graphs which achieve this upper bound. For an arbitrary graph G we furnish two conditions which are necessary if 7(G) + A(G) = n and are sufficien...

There are infinite sequences of graphs {Gn} where |Gn| = n such that the minimal dominating sets for H × Gi fall into predictable patterns, in light of which γ(Gn × H) may be nearly linear in n; the coefficient of linearity may be regarded as the average density of the dominating set in the H-fibers of the product. The specific cases where the sequence {Gn} consists of cycles or path is explore...

We show that the number of minimal dominating sets in a graph on n vertices is at most 1.7697, thus improving on the trivial O(2n/√n) bound. Our result makes use of the measure and conquer technique from exact algorithms, and can be easily turned into an O(1.7697) listing algorithm. Based on this result, we derive an O(2.8805n) algorithm for the domatic number problem, and an O(1.5780) algorith...

Besides the classical chromatic and achromatic numbers of a graph related to minimum or minimal vertex partitions into independent sets, the b-chromatic number was introduced in 1998 thanks to an alternative definition of the minimality of such partitions. When independent sets are replaced by dominating sets, the parameters corresponding to the chromatic and achromatic numbers are the domatic ...

This paper studies the choosability of signed planar graphs. We prove that every signed planar graph is 5-choosable and that there is a signed planar graph which is not 4-choosable while the unsigned graph is 4-choosable. For each k ∈ {3, 4, 5, 6}, every signed planar graph without circuits of length k is 4-choosable. Furthermore, every signed planar graph without circuits of length 3 and of le...

A signed graph is a graph in which every edge is designated to be either positive or negative; it is balanced if every cycle contains an even number of negative edges. A marked signed graph is a signed graph each vertex of which is designated to be positive or negative, and it is consistent if every cycle in the signed graph possesses an even number of negative vertices. Signed line graph L(S) ...