Locally Irregular-Connected Graphs

A graph G is locally irregular if every two adjacent vertices have distinct degrees and is locally irregular-connected if for every two vertices u and v of G, there is a locally irregular u−v path in G. For a finite set S of two or more positive integers with maximum element k, it is known that there exists a graph of order k+1 with degree set S. This result is extended by showing that there is a locally irregular-connected graph of order k+1 with degree set S. Characterizations are established for all locally irregular-connected graphs of cycle rank at most 2. All sets S of positive integers are determined for which there is a locally irregular-connected graph of cycle rank at most 2 with degree set S. The minimum order of a locally irregular-connected graph with a prescribed degree set is determined as well. Other results and open questions are also presented.