No abstract

For any two vertices u and v in a connected graph G, a u – v path is a monophonic path if it contains no chords, and the monophonic distance dm(u, v) from u to v is defined as the length of a longest u – v monophonic path in G. A u – v monophonic path of length dm(u, v) is called a u – v monophonic. The monophonic eccentricity em(v) of a vertex v in G is the maximum monophonic distance from v to a vertex of G. The monophonic radius rad m G of G is the minimum monophonic eccentricity among the vertices of G, while the monophonic diameter diam m G of G is the maximum monophonic eccentricity among the vertices of G. It is shown that rad m G ≤ diam m G for every connected graph G and that every pair a, b of positive integers with a ≤ b is realizable as the monophonic radius and monophonic diameter of some connected graph. Also, for any three positive integers a, b and c with 3 ≤ a ≤ b ≤ c, there is a connected graph G such that rad G = a, rad m G = b and rad DG = c; and for any three positive integers a, b and c with 5 ≤ a ≤ b ≤ c, there is a connected graph G such that diam G = a, diam m G = b and diam D G = c, where rad G, diam G, rad DG and diam D G denote the radius, diameter, detour radius and detour diameter, respectively. The monophonic center of G is the subgraph induced by the vertices of G having monophonic eccentricity rad m G and it is shown that every graph is the monophonic center of some connected graph and also that the monophonic center Cm(G) of every connected graph G is a subgraph of some block of G.

For a connected graph G = (V, E) of order at least two, a set S of vertices of G is a monophonic set of G if each vertex v of G lies on an x − y monophonic path for some elements x and y in S. The minimum cardinality of a monophonic set of G is the monophonic number of G, denoted by m(G). Certain general properties satisfied by the monophonic sets are studied. Graphs G of order p with m(G) = 2 or p or p − 1 are characterized. For every pair a, b of positive integers with 2 ≤ a ≤ b, there is a connected graph G with m(G) = a and g(G) = b, where g(G) is the geodetic number of G. Also we study how the monophonic number of a graph is affected when pendant edges are added to the graph.

For two vertices u and v in a graph G = (V, E), the detour distance D (u, v) is the length of a longest u-v path in G. A u-v path of length D(u, v) is called a u-v detour. A set S ⊆ V is called a detour set of G if every vertex in G lies on a detour joining a pair of vertices of S. The detour number dn(G) of G is the minimum order of its detour sets and any detour set of order dn(G) is a detour basis of G. A set S ⊆ V is called a connected detour set of G if S is detour set of G and the subgraph G[S] induced by S is connected. The connected detour number cdn(G) of G is the minimum order of its connected detour sets and any connected detour set of order cdn(G) is called a connected detour basis of G. A subset T of a connected detour basis S is called a forcing subset for S if S is the unique connected detour basis containing T . A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing connected detour number of S, denoted by fcdn(S), is the cardinality of a minimum forcing subset for S. The forcing connected detour number of G, denoted by fcdn(G), is fcdn(G) = min{fcdn(S)}, where the minimum is taken over all connected detour bases S in G. The forcing connected detour numbers of certain standard graphs are obtained. It is shown that for each pair a, b of integers with 0 ≤ a < b and b ≥ 3, there is a connected graph G with fcdn(G) = a and cdn(G) = b.

For vertices x and y in a connected graph G, the detour distance D(x, y) is the length of a longest x − y path in G. An x − y path of length D(x, y) is an x − y detour. The closed detour interval I D [x, y] consists of x, y, and all vertices lying on some x − y detour of G; while for S ⊆ V (G), I D [S] =