Semiharmonic graphs with fixed cyclomatic number

Let the trunk of a graph G be the graph obtained by removing all leaves of G. We prove that, for every integer c greater than or equal to 2, there are at most finitely many trunks of semiharmonic graphs with cyclomatic number c-in contrast to the fact established by the last two of the present authors in their paper Semiharmonic Bicyclic Graphs (this journal) that there are infinitely many connected semiharmonic graphs with given cyclomatic number. Further, we prove that there are at most finitely many semiharmonic but not almost semiregular graphs with cyclomatic number c. (C) 2004 Elsevier Ltd. All rights reserved.