Proper Total Domination in Generalized Fans and Wheels

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to (totally dominated by) some vertex of S. The number of vertices in S that totally dominate a vertex v of G is denoted by σS(v). A total dominating set S is a pt-dominating set (or pt-dominating set) if σS(u)≠σS(v) for every two adjacent vertices u and v of G. The pt-domination number γpt(G) of a graph G is the minimum cardinality of a pt-dominating set in G. Two well-known classes of highly symmetric graphs are investigated here, namely fans and wheels. For n≥2, the fan Fn is the join Pn∨K1 of the path Pn of order n and the complete graph K1 of order 1, while for n≥3, the wheel Wn is the join Cn∨K1 of the cycle Cn of order n and K1. For a positive integer t, the general fan Fn,t is the graph Pn∨K¯t and the general wheel Wn,t is the graph Cn∨K¯t where K¯t is the empty graph of order t. All fans, wheels, general fans and general wheels are determined that possess pt-dominating sets. Furthermore, pt-domination numbers of all these graphs are determined as well.