A study of inverse, mixed block domination and related concepts in graphs

Abstract

It is proved that every graph is a block tree. Varieties of block- degrees and block-regular graphs have been defined. Expressions for the sum of block degrees are obtained. Several bounds for number of blocks and number of cutvertices in a graph are obtained. Using block distance we defined the central tendencies of a block, like B-radius and B-diameter of a graph. It is proved that block center of any graph is B-complete. Expressions for number of edges in the newly defined graphs are derived. Various block domination parameters are defined. A vertex vand#61646;V and a block band#61646;B (G) is said to block dominate (b-dominate) each other if v is incident on the block b. A set D and#61645; V is said to be a vertex-block dominating set (VBD set) if every block in G is b-dominated by some vertex in D. The vertex - block domination number and#61543;vb = and#61543;vb(G) is the cardinality of a minimum block dominating set of G. We initiate a study of these new varieties of mixed block domination in graphs. We define another parameter called inverse independence number of a graph. Extending the same concept for edge independent sets inverse matching number of a graph is defined. Using minimum clique number , independent domination number of complement of any graph G is obtained. Finally in the last section, various inverse block domination parameters are defined and initiated the study of properties of inverse block domination.