Triangles and (Total) Domination in Subcubic Graphs

A set S of vertices in a graph G is a dominating set if every vertex not in S is adjacent to a vertex in S. If, in addition, every vertex in S is adjacent to some other vertex in S, then S is a total dominating set. The domination number gamma(G) of G is the minimum cardinality of a dominating set in G, while the total domination number gamma(t)(G) of G is the minimum cardinality of a total dominating set in G. The matching number alpha'(G) is the cardinality of a maximum matching in G. A claw-free (resp., diamond-free) graph is a graph that does not contain K-1,K-3 (resp., K-4 - e) as an induced subgraph. We characterize the connected claw-free cubic graphs with maximum possible domination number. We prove that if G is a connected graph of order n with minimum degree delta(G) = 2 and maximum degree Delta(G) = 3, then alpha'(G) >= 2/5n, and we characterize the (infinite) family of extremal graphs. Using this matching result, we prove that if G is a diamond-free special subcubic graph of order n, then gamma(t)(G) <= 2/5n, where a special subcubic graph is a graph G with 2 <= delta(G) and Delta(G) = 3 that satisfies the property that (i) every vertex of G belongs to a triangle and (ii) every triangle contains at most one vertex of degree 2 in G. This result generalizes known results.