VERTEX-TO-CLIQUE MONOPHONIC DISTANCE IN GRAPHS

Let C be a clique and u a vertex in a connected graph G. A vertex-to-clique u - C path P is a u - v path, where v is a vertex in C such that P contains no vertices of C other than v and the u - C path P is said to be u - C monophonic path if P contains no chords in G. The vertex-to-clique monophonic distance d(m)(u, C) is the length of a longest u - C monophonic path in G. A u - C monophonic path of length d(m)(u, C) is called a vertex-to-clique u - C monophonic. The vertex-to-clique monophonic eccentricity e(m1) (u) of a vertex u in G is the maximum vertex-to-clique monophonic distance from u to a clique C is an element of zeta in G, where zeta is the set of all cliques in G. The vertex-to-clique monophonic radius R-m1 of G is the minimum vertex to-clique monophonic eccentricity among the vertices of G, while the vertex-to-clique monophonic diameter D-m1 of G is the maximum vertex-to-clique monophonic eccentricity among the vertices of G. It is shown that R-m1 <= D-m1 for every connected graph G and that every two positive integers a and 6 with 2 <= a <= 6 are realizable as the vertex-to-clique monophonic radius and the vertex-to-clique monophonic diameter, respectively, of some connected graph. The vertex-to-clique monophonic center C-m1 (G) is the subgraph induced by the set of all vertices having minimum vertex-to-clique monophonic eccentricity and the vertex-to-clique monophonic periphery P-m1 (C) is the subgraph induced by the set of all vertices having maximum vertex-to-clique monophonic eccentricity. It is shown that the vertex-to-clique monophonic center of every connected graph G lies in a single block of G.