On total covers of graphs

Mixed Roman Domination in Graphs

“…For every vertex v ∈ V , the open neighborhood N (v) is the set {u ∈ V | uv ∈ E}, and the closed neighborhood of v is the set [7,14,18]. For a mixed dominating set S ⊆ V ∪ E, every element in (V ∪ E)\S is adjacent or incident to an element in S. The mixed domination number, denoted γ * (G), of G is the minimum cardinality of any mixed dominating set of G. A mixed dominating set is also called a total cover in [2,3,10,11].…”

Upper and Lower Bounds on Approximating Weighted Mixed Domination

“…They used it to minimize the number of phase measurement units (PMUs) needed to be placed and maintain the ability of monitoring the entire system. We can see that Mixed Domination has drawn certain attention since its introduction [15,16,3,23,12].…”

An efficient algorithm for Mixed domination on Generalized Series-Parallel Graphs

“…The minimum cardinality among all mixed dominating sets in G is denoted by γ m (G). In [1], Alavi et al showed that this number is bounded from above by n/2 for a connected graph G of order n. They illustrated some extremal cases and gave some properties for connected graphs which have a total covering number equal to n/2 in [3]. In [4], Majumdar showed that the problem of finding γ m (G) is NP-complete for general graphs.…”

“…j and j r ) are determined as follows (they are also shown in Table 2). Add (5, 6), (6,5), (6,6) to list 27: end procedure Case 1. i = 1 implies i = i r = 1, (3,6), (4,4), (4,5), (4, 6)} or (i r , i ) ∈ { (3,4), (3,6), (4,4), (4,5), (4, 6)} Case 5. i = 5 implies i = i r = 5. Case 6. i = 6 implies (i , i r ) ∈ {(5, 6), (6,5), (6,6)}.…”