2 research outputs found
Monophonic Distance in Graphs
For any two vertices u and v in a connected graph G, a u β v path is a monophonic path if it contains no chords, and the monophonic distance dm(u, v) is the length of a longest u β v monophonic path in G. For any vertex v in G, the monophonic eccentricity of v is em(v) = max {dm(u, v) : u β V}. The subgraph induced by the vertices of G having minimum monophonic eccentricity is the monophonic center of G, and it is proved that every graph is the monophonic center of some graph. Also it is proved that the monophonic center of every connected graph G lies in some block of G. With regard to convexity, this monophonic distance is the basis of some detour monophonic parameters such as detour monophonic number, upper detour monophonic number, forcing detour monophonic number, etc. The concept of detour monophonic sets and detour monophonic numbers by fixing a vertex of a graph would be introduced and discussed. Various interesting results based on these parameters are also discussed in this chapter
Median of a graph with respect to edges
For any vertex v and any edge e in a non-trivial connected graph G, the distance sum d(v) of v is , the vertex-to-edge distance sum dβ(v) of v is , the edge-to-vertex distance sum dβ(e) of e is and the edge-to-edge distance sum dβ(e) of e is . The set M(G) of all vertices v for which d(v) is minimum is the median of G; the set Mβ(G) of all vertices v for which dβ(v) is minimum is the vertex-to-edge median of G; the set Mβ(G) of all edges e for which dβ(e) is minimum is the edge-to-vertex median of G; and the set Mβ(G) of all edges e for which dβ(e) is minimum is the edge-to-edge median of G. We determine these medians for some classes of graphs. We prove that the edge-to-edge median of a graph is the same as the median of its line graph. It is shown that the center and the median; the vertex-to-edge center and the vertex-to-edge median; the edge-to-vertex center and the edge-to-vertex median; and the edge-to-edge center and the edge-to-edge median of a graph are not only different but can be arbitrarily far apart