# The connected forcing connected vertex detour number of a graph

Discussiones Mathematicae Graph Theory (2011)

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

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topA.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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