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The vertex detour hull number of a graph

100%
Santhakumaran A., Ullas Chandran S.
Discussiones Mathematicae Graph Theory
|
2012
|
tom 32
|
nr 2
321-330
EN
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), $I_D[S] = ⋃_{x,y ∈ S} I_D[x,y]$. A set S of vertices is a detour convex set if $I_D[S] = S$. The detour convex hull $[S]_D$ is the smallest detour convex set containing S. The detour hull number dh(G) is the minimum cardinality among subsets S of V(G) with $[S]_D = V(G)$. Let x be any vertex in a connected graph G. For a vertex y in G, denoted by $I_D[y]^x$, the set of all vertices distinct from x that lie on some x - y detour of G; while for S ⊆ V(G), $I_D[S]^x = ⋃_{y ∈ S} I_D[y]^x$. For x ∉ S, S is an x-detour convex set if $I_D[S]^x = S$. The x-detour convex hull of S, $[S]^x_D$ is the smallest x-detour convex set containing S. A set S is an x-detour hull set if $[S]^x_D = V(G) -{x}$ and the minimum cardinality of x-detour hull sets is the x-detour hull number dhₓ(G) of G. For x ∉ S, S is an x-detour set of G if $I_D[S]^x = V(G) - {x}$ and the minimum cardinality of x-detour sets is the x-detour number dₓ(G) of G. Certain general properties of the x-detour hull number of a graph are studied. It is shown that for each pair of positive integers a,b with 2 ≤ a ≤ b+1, there exist a connected graph G and a vertex x such that dh(G) = a and dhₓ(G) = b. It is proved that every two integers a and b with 1 ≤ a ≤ b, are realizable as the x-detour hull number and the x-detour number respectively. Also, it is shown that for integers a,b and n with 1 ≤ a ≤ n -b and b ≥ 3, there exist a connected graph G of order n and a vertex x such that dhₓ(G) = a and the detour eccentricity of x, $e_D(x) = b$. We determine bounds for dhₓ(G) and characterize graphs G which realize these bounds.
2
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The hull number of strong product graphs

100%
Santhakumaran A., Ullas Chandran S.
Discussiones Mathematicae Graph Theory
|
2011
|
tom 31
|
nr 3
493-507
EN
For a connected graph G with at least two vertices and S a subset of vertices, the convex hull $[S]_G$ is the smallest convex set containing S. The hull number h(G) is the minimum cardinality among the subsets S of V(G) with $[S]_G = V(G)$. Upper bound for the hull number of strong product G ⊠ H of two graphs G and H is obtainted. Improved upper bounds are obtained for some class of strong product graphs. Exact values for the hull number of some special classes of strong product graphs are obtained. Graphs G and H for which h(G⊠ H) = h(G)h(H) are characterized.
3
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The edge geodetic number and Cartesian product of graphs

100%
Santhakumaran A., Ullas Chandran S.
Discussiones Mathematicae Graph Theory
|
2010
|
tom 30
|
nr 1
55-73
EN
For a nontrivial connected graph G = (V(G),E(G)), a set S⊆ V(G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g₁(G) of G is the minimum order of its edge geodetic sets. Bounds for the edge geodetic number of Cartesian product graphs are proved and improved upper bounds are determined for a special class of graphs. Exact values of the edge geodetic number of Cartesian product are obtained for several classes of graphs. Also we obtain a necessary condition of G for which g₁(G ☐ K₂) = g₁(G).
4
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The geodetic number of strong product graphs

100%
Santhakumaran A., Ullas Chandran S.
Discussiones Mathematicae Graph Theory
|
2010
|
tom 30
|
nr 4
687-700
EN
For two vertices u and v of a connected graph G, the set $I_G[u,v]$ consists of all those vertices lying on u-v geodesics in G. Given a set S of vertices of G, the union of all sets $I_G[u,v]$ for u,v ∈ S is denoted by $I_G[S]$. A set S ⊆ V(G) is a geodetic set if $I_G[S] = V(G)$ and the minimum cardinality of a geodetic set is its geodetic number g(G) of G. Bounds for the geodetic number of strong product graphs are obtainted and for several classes improved bounds and exact values are obtained.
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