Cycle Domination, Independence and Irredundance in graphs

Abstract

A set S of vertices in a graph G = (V, E) is called cycle independent if the induced subgraph S is acyclic, and called oddcycle indepdendet if S is bipartite. A set S is cycle dominating (resp. odd-cycle dominating) if for every vertex u ∈ V \ S there exists a vertex v ∈ S such that u and v are contained in a (resp. odd cycle) cycle in S \{u} . A set S is cycle irredundant (resp. odd-cycle irredundant) if for every vertex v ∈ S there exists a vertex u ∈ V \ S such that u and v are in a (resp. odd cycle) cycle of S \ {u} , but u is not in a cycle of S ∪ {u} \ {v} . In this paper we present these new concepts, which relate in a natural way to independence, domination and irredundance in graphs. In particular, we construct analogs to the domination inequality chain for these new concepts.

Domination chain: Characterisation, classical complexity, parameterised complexity and approximability

“…For the ease of reference, the six parameters of the domination chain are defined in Table 1. Over time, other parameters were added to the domination chain (see, for example, [51]).…”