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Axiomatic characterizations of Ptolemaic and chordal graphs

Changat Manoj, Sheela Lekshmi Kamal K., Narasimha-Shenoi Prasanth G.

Opuscula Mathematica

2023

Vol. 43, no. 3

393--407

The interval function and the induced path function are two well studied class of set functions of a connected graph having interesting properties and applications to convexity, metric graph theory. Both these functions can be framed as special instances of a general set function termed as a transit function defined on the Cartesian product of a non-empty set V to the power set of V satisfying the expansive, symmetric and idempotent axioms. In this paper, we propose a set of independent first order betweenness axioms on an arbitrary transit function and provide characterization of the interval function of Ptolemaic graphs and the induced path function of chordal graphs in terms of an arbitrary transit function. This in turn gives new characterizations of the Ptolemaic and chordal graphs.

On the General Position Number of Complementary Prisms

Neethu P. K., Chandran S. V. Ullas, Changat Manoj, Klavžar Sandi

Fundamenta Informaticae

2021

Vol. 178, nr 3

267--281

The general position number gp(G ) of a graph G is the cardinality of a largest set of vertices S such that no element of S lies on a geodesic between two other elements of S. The complementary prism G G ¯ of G is the graph formed from the disjoint union of G and its complement G ¯ by adding the edges of a perfect matching between them. It is proved that gp(G G ¯ ) ≤ n (G ) + 1 if G is connected and gp(G G ¯ ) ≤ n (G ) if G is disconnected. Graphs G for which gp(G G ¯ ) = n (G ) + 1 holds, provided that both G and G ¯ are connected, are characterized. A sharp lower bound on gp(G G ¯ ) is proved. If G is a connected bipartite graph or a split graph then gp(G G ¯ ) ∈ {n (G ), n (G )+1}. Connected bipartite graphs and block graphs for which gp(G G ¯ ) = n (G ) + 1 holds are characterized. A family of block graphs is constructed in which the gp-number of their complementary prisms is arbitrary smaller than their order.