Domination changing and unchanging signed graphs upon the vertex removal

A subset S of V (Σ) is a dominating set of Σ if |N⁺(v) ∩ S| > |N⁻(v) ∩ S| for all v ∈ V − S. This article is to start a study of those signed graphs that are stable and critical in the following way: If the removal of an arbitrary vertex does not change the domination number, the signed graph will be stable. The signed graph, on the other hand, is unstable if an arbitrary vertex is removed and the domination number changes. Specifically, we analyze the change in the domination of the vertex deletion and stable signed graphs.Publisher's Versio

Fractional global domination in graphs

Let G = (V,E) be a graph. A function g:V → [0,1] is called a global dominating function (GDF) of G, if for every v ∈ V, $g(N[v]) = ∑_{u ∈ N[v]}g(u) ≥ 1$ and $g(\overline{N(v)}) = ∑_{u ∉ N(v)}g(u) ≥ 1$. A GDF g of a graph G is called minimal (MGDF) if for all functions f:V → [0,1] such that f ≤ g and f(v) ≠ g(v) for at least one v ∈ V, f is not a GDF. The fractional global domination number $γ_{fg}(G)$ is defined as follows: $γ_{fg}(G)$ = min{|g|:g is an MGDF of G } where $|g| = ∑_{v ∈ V} g(v)$. In this paper we initiate a study of this parameter

Edge open packing sets in graphs

In a graph G = (V, E), two edges e1 and e2 are said to have a common edge if there exists an edge e ∈ E(G) different from e1 and e2 such that e joins a vertex of e1 to a vertex of e2 in G. That is, 〈e1, e, e2〉 is either P4 or K3 in G. A non-empty set D ⊆ E(G) is an edge open packing set of a graph G if no two edges of D have a common edge in G. The maximum cardinality of an edge open packing set is the edge open packing number of G and is denoted by ${\rho }_e^o(G)$. In this paper, we initiate a study on this parameter