Let G be a connected graph. Let f be a proper k -coloring of G and Π = (R_1, R_2, . . . , R_k) bean ordered partition of V (G) into color classes. For any vertex v of G, define the color code c_Π(v) of v with respect to Π to be a k -tuple (d(v, R_1), d(v, R_2), . . . , d(v, R_k)), where d(v, R_i) is the min{d(v, x)|x ∈ R_i}. If distinct vertices have distinct color codes, then we call f a locat...

A k-dominating set is a set D k V such that every vertex i 2 V nD k has at least k i neighbours in D k. The k-domination number k (G) of G is the cardinality of a smallest k-dominating set of G. For k 1 = ::: = kn = 1, k-domination corresponds to the usual concept of domination. Our approach yields an improvement of an upper bound for the domination number found then the notion of k-dominating ...

A k-tuple coloring of a graph G assigns a set of k colors to each vertex of G so that if two vertices are adjacent, the corresponding sets of colors are disjoint. The k-tuple chromatic number of G is the smallest t so that there is such a k-tuple coloring of G using t colors in all. The Kneser graph K(m,n) has as vertices all n-element subsets of the set {1,2, . . . ,m} and an edge between two ...

Let G = (V , E) be a graph. A subset D ⊆ V is a dominating set if every vertex not in D is adjacent to a vertex in D. A dominating set D is called a total dominating set if every vertex in D is adjacent to a vertex in D. The domination (resp. total domination) number of G is the smallest cardinality of a dominating (resp. total dominating) set of G. The bondage (resp. total bondage) number of a...

Domination theory is a well-established topic in graph theory, as well one of the most active research areas. Interest this area partly explained by its diversity applications to real-world problems, such facility location computer and social networks, monitoring communication, coding algorithm design, among others. In last two decades, functions defined on graphs have attracted attention sever...

In this paper we introduce and study a new graph invariant derived from the degree sequence of a graph G, called the sub-k-domination number and denoted subk(G). We show that subk(G) is a computationally efficient sharp lower bound on the k-domination number of G, and improves on several known lower bounds. We also characterize the sub-k-domination numbers of several families of graphs, provide...

‎for any integer $kge 1$‎, ‎a minus $k$-dominating function is a‎ ‎function $f‎ : ‎v (g)rightarrow {-1,0‎, ‎1}$ satisfying $sum_{win‎‎n[v]} f(w)ge k$ for every $vin v(g)$‎, ‎where $n(v) ={u in‎‎v(g)mid uvin e(g)}$ and $n[v] =n(v)cup {v}$‎. ‎the minimum of‎‎the values of $sum_{vin v(g)}f(v)$‎, ‎taken over all minus‎‎$k$-dominating functions $f$‎, ‎is called the minus $k$-domination‎‎number and i...