The Forcing Weak Edge Detour Number of a Graph

The forcing weak edge detour numbers of certain classes of graphs are determined

UNIQUE ECCENTRIC CLIQUE GRAPHS

Let $G$ be a connected graph and $\zeta$ the set of all cliques in $G$. In this paper we introduce the concepts of unique $(\zeta, \zeta)$-eccentric clique graphs and self $(\zeta, \zeta)$-centered graphs. Certain standard classes of graphs are shown to be self $(\zeta, \zeta)$-centered, and we characterize unique $(\zeta, \zeta)$-eccentric clique graphs which are self $(\zeta, \zeta)$-centered

Monophonic Distance in Graphs

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) is the length of a longest u − v monophonic path in G. For any vertex v in G, the monophonic eccentricity of v is em(v) = max {dm(u, v) : u ∈ 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

The upper edge-to-vertex detour number of a 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. For subsets A and B of V, the detour distance D(A, B) is defined as D(A, B) = min{D(x, y): x ∈ A, y ∈ B}. A u-v path of length D(A, B) is called an A-B detour joining the sets A, B ⊆ V where u ∈ A and v ∈ B. A vertex x is said to lie on an A-B detour if x is a vertex of an A-B detour. A set S ⊆ E is called an edge-to-vertex detour set if every vertex of G is incident with an edge of S or lies on a detour joining a pair of edges of S. The edge-to-vertex detour number dn₂(G) of G is the minimum order of its edge-to-vertex detour sets and any edge-to-vertex detour set of order dn₂(G) is an edge-to-vertex detour basis of G. An edge-to-vertex detour set S in a connected graph G is called a minimal edge-to-vertex detour set of G if no proper subset of S is an edge-to-vertex detour set of G. The upper edge-to-vertex detour number, dn₂⁺(G) of G is the maximum cardinality of a minimal edge-to-vertex detour set of G. The upper edge-to-vertex detour numbers of certain standard graphs are obtained. It is shown that for every pair a, b of integers with 2 ≤ a ≤ b, there exists a connected graph G with dn2(G) = a and dn₂⁺(G) = b

The detour hull number of a graph

For vertices u and v in a connected graph G = (V, E), the set ID[u, v] consists of all those vertices lying on a u−v longest path in G. Given a set S of vertices of G, the union of all sets ID[u, v] for u, v ∈ S, is denoted by ID[S]. A set S is a detour convex set if ID[S] = S. The detour convex hull [S]D of S in G is the smallest detour convex set containing S. The detour hull number dh(G) is the minimum cardinality among the subsets S of V with [S]D = V. A set S of vertices is called a detour set if ID[S] = V. The minimum cardinality of a detour set is the detour number dn(G) of G. A vertex x in G is a detour extreme vertex if it is an initial or terminal vertex of any detour containing x. Certain general properties of these concepts are studied. It is shown that for each pair of positive integers r and s, there is a connected graph G with r detour extreme vertices, each of degree s. Also, it is proved that every two integers a and b with 2 ≤ a ≤ b are realizable as the detour hull number and the detour number respectively, of some graph. For each triple D, k and n of positive integers with 2 ≤ k ≤ n − D + 1 and D ≥ 2, there is a connected graph of order n, detour diameter D and detour hull number k. Bounds for the detour hull number of a graph are obtained. It is proved that dn(G) = dh(G) for a connected graph G with detour diameter at most 4. Also, it is proved that for positive integers a, b and k ≥ 2 with a < b ≤ 2a, there exists a connected graph G with detour radius a, detour diameter b and detour hull number k. Graphs G for which dh(G) = n − 1 or dh(G) = n − 2 are characterized

THE UPPER OPEN GEODETIC NUMBER OF A GRAPH

For a connected graph G of order n, a set S of vertices of G is a geodetic set of G if each vertex  n of G lies on a x-y geodesic for some elements x and y in S. The minimum cardinality of a geodetic set of G is defined as the geodetic number of G, denoted by g(G). A geodetic set of cardinality g(G) is called a g-set of G. A set S of  vertices of  a connected graph G is an open geodetic set of G if for each vertex n in G, either n is an extreme vertex of G and n  S; or n is an internal vertex of an x-y geodesic for some x,yS. An open geodetic set of minimum cardinality is a minimum open geodetic set and this cardinality is the open geodetic number, og(G). An open geodetic set S in a connected graph G is called a minimal open geodetic set if no proper subset of S is an open geodetic set of G. The upper open geodetic number og+(G) of G is the maximum cardinality of a minimal open geodetic set of G. It is shown that, for a connected graph G of order n, og(G)=n, if and only if og+(G)=n, and also that og(G)=3 if any only if og+(G)=3. It is shown that for positive integers a and b with 4 ≤ a ≤ b, there exists a connected graph G with og(G) =a and og+(G)=b. Also, it is shown that for positive integers a, b, c with 4 ≤ a ≤ b ≤ c and b ≤  3a, there exists a connected graph G with g(G)=a, og(G)=b and og+(G)= c

On edge detour graphs

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 an edge detour set if every edge in G lies on a detour joining a pair of vertices of S. The edge detour number dn₁(G) of G is the minimum order of its edge detour sets and any edge detour set of order dn₁(G) is an edge detour basis of G. A connected graph G is called an edge detour graph if it has an edge detour set. It is proved that for any non-trivial tree T of order p and detour diameter D, dn₁(T) ≤ p-D+1 and dn₁(T) = p-D+1 if and only if T is a caterpillar. We show that for each triple D, k, p of integers with 3 ≤ k ≤ p-D+1 and D ≥ 4, there is an edge detour graph G of order p with detour diameter D and dn₁(G) = k. We also show that for any three positive integers R, D, k with k ≥ 3 and R < D ≤ 2R, there is an edge detour graph G with detour radius R, detour diameter D and dn₁(G) = k. Edge detour graphs G with detour diameter D ≤ 4 are characterized when dn₁(G) = p-2 or dn₁(G) = p-1