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.
This paper considers distributed storage systems (DSSs) from a graph theoretic perspective. A DSS is constructed by means of the path decomposition of a 3-regular graph into P 4 paths. The paths represent the disks of the DSS and the edges of the graph act as the blocks of storage. We deduce the properties of the DSS from a related graph and show their optimality.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.