Capacitated Domination generalizes the classic Dominating Set problem by specifying for each vertex a required demand and an available capacity for covering demand in its closed neighborhood. The objective is to find a minimum-sized set of vertices that can cover all of the graph’s demand without exceeding any of the capacities. In this paper we look specifically at domination with hard-capacit...

A graph is said to be well-dominated if all its minimal dominating sets are of the same size. In this work, we introduce the notion of an irreducible dominating set, a variant of dominating set generalizing both minimal dominating sets and minimal total dominating sets. Based on this notion, we characterize the family of minimal dominating sets in a lexicographic product of two graphs and deriv...

Let G = (V,E) be a simple graph. A set S ⊆ V is a dominating set of G, if every vertex in V \S is adjacent to at least one vertex in S. Let C n be the family of dominating sets of a cycle Cn with cardinality i, and let d(Cn, i) = |C i n|. In this paper, we construct C i n, and obtain a recursive formula for d(Cn, i). Using this recursive formula, we consider the polynomial D(Cn, x) = ∑n i=⌈n 3 ...

Let G = (V,E) be a finite undirected graph without loops and multiple edges. A subset M ⊆ E of edges is a dominating induced matching (d.i.m.) in G if every edge in E is intersected by exactly one edge of M . In particular, this means that M is an induced matching, and every edge not in M shares exactly one vertex with an edge in M . Clearly, not every graph has a d.i.m. The Dominating Induced ...

Let G = (V,E) be a finite undirected graph without loops and multiple edges. An edge set E ⊆ E is a dominating induced matching (d.i.m.) in G if every edge in E is intersected by exactly one edge of E. In particular, this means that E is an induced matching, and every edge not in E shares exactly one vertex with an edge in E. Clearly, not every graph has a d.i.m. The Dominating Induced Matching...

A Roman dominating function of a graph G = (V, E) is a function f : V → {0, 1, 2} such that every vertex x with f (x) = 0 is adjacent to at least one vertex y with f (y) = 2. The weight of a Roman dominating function is defined to be f (V ) = ∑ x∈V f (x), and the minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. In this paper we first answer ...

A set S ⊆ V is a dominating set of a graph G = (V, E) if each vertex in V is either in S or is adjacent to a vertex in S. A vertex is said to dominate itself and all its neighbors. The domination number γ (G) is the minimum cardinality of a dominating set of G. A set S ⊆ V is a total dominating set of a graph G = (V,E) if each vertex in V is adjacent to a vertex in S. The total domination numbe...

In this paper, we solve a problem by Glover and Punnen (1997) from the context of domination analysis, where the performance of a heuristic algorithm is rated by the number of solutions that are not better than the solution found by the algorithm, rather than by the relative performance compared to the optimal value. In particular, we show that for the Asymmetric Traveling Salesman Problem (ATS...

Given a graph G = (V, E) of order n and an n-dimensional non-negative vector d = (d(1), d(2), . . . , d(n)), called demand vector, the vector domination (resp., total vector domination) is the problem of finding a minimum S ⊆ V such that every vertex v in V \S (resp., in V ) has at least d(v) neighbors in S. The (total) vector domination is a generalization of many dominating set type problems,...