Monophonic-triangular Distance in Graphs

Abstract

A path u1, u2, ..., un in a connected graph G such that for i, j with j ≥ i + 3, there does not exist an edge uiuj , is called a monophonic-triangular path or mt-path. The monophonic-triangular distance or mt-distance dmt(u, v) from u to v is defined as the length of a longest u−v mt-path in G. The mt-eccentricity emt(v) of a vertex v in G is defined as the maximum mt-distance between v and other vertices in G. The mt-radius radmt(G) is defined as the minimum mt-eccentricity among the vertices of G and the mt-diameter diammt(G) is defined as the maximum mt-eccentricity among the vertices of G. It is shown that radmt(G) ≤ diammt(G) for every connected graph G. Some realization and characterization results are given based on mt-radius, mt-diameter, mt-center and mt-periphery of a connected graph.