# The connected forcing connected vertex detour number of a graph

• Volume: 31, Issue: 3, page 461-473
• ISSN: 2083-5892

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## Abstract

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For any vertex x in a connected graph G of order p ≥ 2, a set S of vertices of V is an x-detour set of G if each vertex v in G lies on an x-y detour for some element y in S. A connected x-detour set of G is an x-detour set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected x-detour set of G is the connected x-detour number of G and is denoted by cdₓ(G). For a minimum connected x-detour set Sₓ of G, a subset T ⊆ Sₓ is called a connected x-forcing subset for Sₓ if the induced subgraph G[T] is connected and Sₓ is the unique minimum connected x-detour set containing T. A connected x-forcing subset for Sₓ of minimum cardinality is a minimum connected x-forcing subset of Sₓ. The connected forcing connected x-detour number of Sₓ, denoted by $c{f}_{cdx}\left(Sₓ\right)$, is the cardinality of a minimum connected x-forcing subset for Sₓ. The connected forcing connected x-detour number of G is $c{f}_{cdx}\left(G\right)=minc{f}_{cdx}\left(Sₓ\right)$, where the minimum is taken over all minimum connected x-detour sets Sₓ in G. Certain general properties satisfied by connected x-forcing sets are studied. The connected forcing connected vertex detour numbers of some standard graphs are determined. It is shown that for positive integers a, b, c and d with 2 ≤ a < b ≤ c ≤ d, there exists a connected graph G such that the forcing connected x-detour number is a, connected forcing connected x-detour number is b, connected x-detour number is c and upper connected x-detour number is d, where x is a vertex of G.

## How to cite

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A.P. Santhakumaran, and P. Titus. "The connected forcing connected vertex detour number of a graph." Discussiones Mathematicae Graph Theory 31.3 (2011): 461-473. <http://eudml.org/doc/271053>.

@article{A2011,
abstract = {For any vertex x in a connected graph G of order p ≥ 2, a set S of vertices of V is an x-detour set of G if each vertex v in G lies on an x-y detour for some element y in S. A connected x-detour set of G is an x-detour set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected x-detour set of G is the connected x-detour number of G and is denoted by cdₓ(G). For a minimum connected x-detour set Sₓ of G, a subset T ⊆ Sₓ is called a connected x-forcing subset for Sₓ if the induced subgraph G[T] is connected and Sₓ is the unique minimum connected x-detour set containing T. A connected x-forcing subset for Sₓ of minimum cardinality is a minimum connected x-forcing subset of Sₓ. The connected forcing connected x-detour number of Sₓ, denoted by $cf_\{cdx\}(Sₓ)$, is the cardinality of a minimum connected x-forcing subset for Sₓ. The connected forcing connected x-detour number of G is $cf_\{cdx\}(G) = mincf_\{cdx\}(Sₓ)$, where the minimum is taken over all minimum connected x-detour sets Sₓ in G. Certain general properties satisfied by connected x-forcing sets are studied. The connected forcing connected vertex detour numbers of some standard graphs are determined. It is shown that for positive integers a, b, c and d with 2 ≤ a < b ≤ c ≤ d, there exists a connected graph G such that the forcing connected x-detour number is a, connected forcing connected x-detour number is b, connected x-detour number is c and upper connected x-detour number is d, where x is a vertex of G.},
author = {A.P. Santhakumaran, P. Titus},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {vertex detour number; connected vertex detour number; upper connected vertex detour number; forcing connected vertex detour number; connected forcing connected vertex detour number},
language = {eng},
number = {3},
pages = {461-473},
title = {The connected forcing connected vertex detour number of a graph},
url = {http://eudml.org/doc/271053},
volume = {31},
year = {2011},
}

TY - JOUR
AU - A.P. Santhakumaran
AU - P. Titus
TI - The connected forcing connected vertex detour number of a graph
JO - Discussiones Mathematicae Graph Theory
PY - 2011
VL - 31
IS - 3
SP - 461
EP - 473
AB - For any vertex x in a connected graph G of order p ≥ 2, a set S of vertices of V is an x-detour set of G if each vertex v in G lies on an x-y detour for some element y in S. A connected x-detour set of G is an x-detour set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected x-detour set of G is the connected x-detour number of G and is denoted by cdₓ(G). For a minimum connected x-detour set Sₓ of G, a subset T ⊆ Sₓ is called a connected x-forcing subset for Sₓ if the induced subgraph G[T] is connected and Sₓ is the unique minimum connected x-detour set containing T. A connected x-forcing subset for Sₓ of minimum cardinality is a minimum connected x-forcing subset of Sₓ. The connected forcing connected x-detour number of Sₓ, denoted by $cf_{cdx}(Sₓ)$, is the cardinality of a minimum connected x-forcing subset for Sₓ. The connected forcing connected x-detour number of G is $cf_{cdx}(G) = mincf_{cdx}(Sₓ)$, where the minimum is taken over all minimum connected x-detour sets Sₓ in G. Certain general properties satisfied by connected x-forcing sets are studied. The connected forcing connected vertex detour numbers of some standard graphs are determined. It is shown that for positive integers a, b, c and d with 2 ≤ a < b ≤ c ≤ d, there exists a connected graph G such that the forcing connected x-detour number is a, connected forcing connected x-detour number is b, connected x-detour number is c and upper connected x-detour number is d, where x is a vertex of G.
LA - eng
KW - vertex detour number; connected vertex detour number; upper connected vertex detour number; forcing connected vertex detour number; connected forcing connected vertex detour number
UR - http://eudml.org/doc/271053
ER -

## References

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14. [14] A.P. Santhakumaran and P. Titus, The Upper Connected Vertex Detour Number of a Graph, communicated. Zbl1289.05142

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