The domination and independent domination numbers of some families of snarks

A dominating set of a graph G is a set S ⊆ V (G) such that every vertex in V (G) either belongs to S or is adjacent to some vertex in S. The domination number is the minimum cardinality of a dominating set of G. An independent dominating set of G is a dominating set that is also independent. The minimum cardinality of an independent dominating set of G is the independent domination number of G. Given the computational complexity of these problems, extensive research has been done on finding bounds or determining these parameters for classes of graphs, especially cubic graphs. Furthermore, determining how far apart these parameters are is also a challenging problem. In this work, we establish some bounds for the domination number and the independent domination number for families of cubic graphs, in particular for Generalized Blanuša Snarks and for two families ofLoupekine Snarks known as LP0-snarks and LP1-snarks. We also show that the parameters are equal for these graphs and conjecture that this equality holds for every snark.