The Private Neighbor Cube

Abstract

Let S be a set of vertices in a graph G (F, E). The authors state that a vertex u in S has a private neighbor (relative to S) if either u is not adjacent to any vertex in S or u is adjacent to a vertex w that is not adjacent to any other vertex in S. Based on the notion of private neighbors, a set of eight graph theoretic parameters can be defined whose inequality relationships can be described by a three-dimensional cube. Most of these parameters have already been studied independently. This paper unifies this study and helps to form a cohesive theory of private neighbors in graphs. Theoretical and algorithmic properties of this private neighbor cube are investigated, and many open questions are raised.

CO-irredundant Ramsey numbers for graphs

“…The set X is called CO-irredundant (abbreviation X is a COirredundant set) if for all x P X The CO in the name denotes the fact that the neighborhoods in the de®nition (2) are Closed and Open, respectively. CO-irredundant sets are not yet wellstudied but they are mentioned brie¯y in [12,13,15]. We observe that COirredundance is a hereditary property.…”

Breaking the 2n-barrier for Irredundance: Two lines of attack

“…There are also some applications of irredundant sets in combinatorial optimization, e.g., locating senders in broadcast and packet radio networks [16]. We mention that determining the lower and upper irredundance numbers are NP-hard problems [24,29]. Computing ir(G) remains NP-hard on bipartite graphs [29], while here IR(G) can be computed in polynomial time using the equality IR(G) = α(G) [14].…”

“…An estimated 100 research papers [22] have been published on the properties of irredundant sets in graphs, e.g., [2,[8][9][10]15,12,20,21,24,29,30]. Now, if D ⊆ V is an (inclusion-wise) minimal dominating set, i.e., no proper subset of D is dominating, then for every v ∈ D, there is some minimality witness, i.e., a vertex that is only dominated by v. In fact, a set is minimal dominating if and only if it is irredundant and dominating [13].…”

A lower bound for the CO-irredundance number of a graph

“…We abbreviate these notations to COIR and coir whenever possible. Nordhaus-Gaddum type results [7] and NP-completeness results [11] have been established for COIR. A set X ⊆ V is called 1-dependent if every vertex of X has an X-spn or an X-ipn.…”

“…CO-irredundant Ramsey numbers were introduced in [6] and also appear in [9,14]. In [2,4,11] CO-irredundance has been embedded in classifications of graph theoretic properties based on the existence of private neighbours.…”