Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 Articles in Press 2025 05 14 Hierarchy of Subfamilies of Ptolemaic Graphs: Axiomatic Characterizations and Interval Functions 14948 10.22049/cco.2025.30144.2335 EN Abdultamim Ahadi Department of Mathematics, University of Kerala, Thiruvananthapuram, Kerala, India Arun Anil Department of Futures Studies, University of Kerala, Thiruvananthapuram, Kerala, India Manoj Changat Department of Futures Studies, University of Kerala, Thiruvananthapuram, Kerala, India Journal Article 2024 11 13 Ptolemaic graphs are precisely the graphs that are both chordal and distance-hereditary. Markenzon et al.~\cite{markenzon} established a hierarchy of Ptolemaic graphs comprising six subfamilies: laminar chordal graphs, block duplicate graphs, block graphs, AC graphs, trees, and paths. In this paper, we present a new proof of the characterization of AC graphs using forbidden induced subgraphs and identify an additional graph class that lies between AC graphs and paths within this hierarchy. <br /><br /><br /><br />The interval function is a well-studied tool in metric graph theory, and the characterization of the interval function of graph families is an interesting problem in metric graph theory having connections to first-order logic. In this paper, we propose a set of independent betweenness axioms for an arbitrary function known as a transit function and provide a characterization of the interval functions corresponding to graphs in the extended hierarchy of subgraphs of Ptolemaic graphs, specifically laminar chordal graphs, block duplicate graphs, and AC graphs. https://comb-opt.azaruniv.ac.ir/article_14948_bfa39cec20b409f6e91dcfb642900eff.pdf