Determination of all graphs whose eccentric graphs are clusters

Akiyama, Jin (Tokyo University of Science)
Kodate, Takako (Tokyo Woman's Christian University)
Matsunaga, Kiyoko (Tokyo University of Science)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 12, No 2 (2024): Electronic Journal of Graph Theory and Applications

A disconnected graph G is called a cluster if G is not union of K2s (1-factor) but union of complete graphs of order at least two. J. Akiyama, K. Ando and D. Avis showed in Lemma 2.1 of [2] that G is equi-eccentric if the eccentric graph Ge is a cluster or pK2. And they also characterized all graphs whose eccentric graphs are complete graphs and pK2 in [2]. In this paper, we determined in Theorem 2 all graphs whose eccentric graphs are clusters, which is an extension of Lemma 2.1 in [2].