# The vertex detour hull number of a graph

• Volume: 32, Issue: 2, page 321-330
• ISSN: 2083-5892

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## Abstract

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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 ID[x,y] consists of x,y, and all vertices lying on some x -y detour of G; while for S ⊆ V(G), ${I}_{D}\left[S\right]={\bigcup }_{x,y\in S}{I}_{D}\left[x,y\right]$. A set S of vertices is a detour convex set if ${I}_{D}\left[S\right]=S$. The detour convex hull ${\left[S\right]}_{D}$ 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 ${\left[S\right]}_{D}=V\left(G\right)$. Let x be any vertex in a connected graph G. For a vertex y in G, denoted by ${I}_{D}{\left[y\right]}^{x}$, the set of all vertices distinct from x that lie on some x - y detour of G; while for S ⊆ V(G), ${I}_{D}{\left[S\right]}^{x}={\bigcup }_{y\in S}{I}_{D}{\left[y\right]}^{x}$. For x ∉ S, S is an x-detour convex set if ${I}_{D}{\left[S\right]}^{x}=S$. The x-detour convex hull of S, ${\left[S\right]}_{D}^{x}$ is the smallest x-detour convex set containing S. A set S is an x-detour hull set if ${\left[S\right]}_{D}^{x}=V\left(G\right)-x$ 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 ${I}_{D}{\left[S\right]}^{x}=V\left(G\right)-x$ 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, ${e}_{D}\left(x\right)=b$. We determine bounds for dhₓ(G) and characterize graphs G which realize these bounds.

## How to cite

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A.P. Santhakumaran, and S.V. Ullas Chandran. "The vertex detour hull number of a graph." Discussiones Mathematicae Graph Theory 32.2 (2012): 321-330. <http://eudml.org/doc/270957>.

@article{A2012,
abstract = {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 ID[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.},
author = {A.P. Santhakumaran, S.V. Ullas Chandran},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {detour; detour number; detour hull number; x-detour number; x-detour hull number; -detour number; -detour hull number},
language = {eng},
number = {2},
pages = {321-330},
title = {The vertex detour hull number of a graph},
url = {http://eudml.org/doc/270957},
volume = {32},
year = {2012},
}

TY - JOUR
AU - A.P. Santhakumaran
AU - S.V. Ullas Chandran
TI - The vertex detour hull number of a graph
JO - Discussiones Mathematicae Graph Theory
PY - 2012
VL - 32
IS - 2
SP - 321
EP - 330
AB - 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 ID[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.
LA - eng
KW - detour; detour number; detour hull number; x-detour number; x-detour hull number; -detour number; -detour hull number
UR - http://eudml.org/doc/270957
ER -

## References

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10. [10] A.P. Santhakumaran and S.V. Ullas Chandran, The detour hull number of a graph, communicated. Zbl1293.05090

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