Total limited packing in graphs

Abstract

We define a k-total limited packing number in a graph, which generalizes the concept of open packing number in graphs, and give several bounds on it. These bounds involve many well known parameters of graphs. Also, we establish a connection among the concepts of tuple domination, tuple total domination and total limited packing that implies some results.

Nordhaus-Gaddum type inequalities for multiple domination and packing parameters in graphs

“…Now let {G, G} = {H, H ′ + K 1 }, where H, H ′ ∈ Π. Then the bounds given in (10) hold with equality, by the proof of Theorem 3.2.…”

“…(10) On the other hand, by the proof of Theorem 3.2 the bounds given in (10) hold with equality if and only if G = H + rK 2 + sK 1 and G = H ′ + r ′ K 2 +s ′ K 1 for some non-negative integers r, s, r ′ , s ′ , where H, H ′ ∈ Π. We assume that the upper bounds (10) hold with equality.…”

“…(10) On the other hand, by the proof of Theorem 3.2 the bounds given in (10) hold with equality if and only if G = H + rK 2 + sK 1 and G = H ′ + r ′ K 2 +s ′ K 1 for some non-negative integers r, s, r ′ , s ′ , where H, H ′ ∈ Π. We assume that the upper bounds (10) hold with equality. Assume first that r > 0 and consider a copy of K 2 on two vertices u and v, as a component of G − H. Then uv / ∈ E(G) and u and v are adjacent to all other n − 2 ≥ 2 vertices of G. This shows that G is connected and δ(G) ≥ 2, a contradiction.…”

“…is the maximum cardinality of a k-limited packing (k-total limited packing) in G. These concepts were introduced and investigated in [6] and [10], respectively. In this paper, we continue the study of these bounds for the above domination parameters.…”

(Open) packing number of some graph products

“…Some applications of this equality for trees can be found in [25] and [28]. As a generalization of the open packing, and a total version of the limited packing, the concept of total limited packing was introduced in [15]. A subset S of the vertices is called a k-total limited packing if the open neighborhood of each vertex has at most k neighbors in S.…”