For an edge xy in a connected graph G of order p ≥ 3, a set S ⊆V(G) is an xy-monophonic set of G if each vertex v ∈V(G) lies on either an x−u monophonic path or a y−u monophonic path for some element u in S. The minimum cardinality of an xymonophonic set of G is defined as the xy-monophonic number of G, denoted by mxy(G). An xy-monophonic set of cardinality mxy(G) is called a mxy-set of G. A connected xy-monophonic set of G is an xy-monophonic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected xy-monophonic set of G is the connected xy-monophonic number of G and is denoted by cmxy(G). A connected xy-monophonic set of cardinality cmxy(G) is called a cmxy-set of G. We determine bounds for it and find the same for some special classes of graphs. If d, n and p ≥ 4 are positive integers such that 2 ≤ d ≤ p−2 and 1 ≤ n ≤ p−1, then there exists a connected graph G of order p, monophonic diameter d and cmxy(G) = n for some edge xy in G. Also, we give some characterization and realization results for the parameter cmxy(G).
Digital Object Identifier (DOI)
Titus, P. and Eldin Vanaja, S.
"Connected Edge Fixed Monophonic Number of a Graph,"
Applied Mathematics & Information Sciences: Vol. 10
, Article 39.
Available at: https://dc.naturalspublishing.com/amis/vol10/iss6/39