ALMOST SELF-CENTERED MEDIAN AND CHORDAL GRAPHS

Almost self-centered graphs were recently introduced as the graphs with exactly two non-central vertices. In this paper we characterize almost self-centered graphs among median graphs and among chordal graphs. In the first case $P_4$ and the graphs obtained from hypercubes by attaching to them a single leaf are the only such graphs. Among chordal graph the variety of almost self-centered graph is much richer, despite the fact that their diameter is at most 3. We also discuss almost self-centered graphs among partial cubes and among $k$-chordal graphs, classes of graphs that generalize median and chordal graphs, respectively. Characterizations of almost self-centered graphs among these two classes seem elusive.