The upper restrained Steiner number of a graph

2019 ◽  
Vol 12 (01) ◽  
pp. 2050004
Author(s):  
J. John ◽  
M. S. Malchijah Raj
Keyword(s):  
Connected Graph ◽  
Proper Subset ◽  
Maximum Cardinality ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Geodetic Number ◽  
Number Formula

A Steiner set [Formula: see text] of a connected graph [Formula: see text] of order [Formula: see text] is a restrained Steiner set if either [Formula: see text] or the subgraph [Formula: see text] has no isolated vertices. The minimum cardinality of a restrained Steiner set of [Formula: see text] is the restrained Steiner number of [Formula: see text], and is denoted by [Formula: see text]. A restrained Steiner set [Formula: see text] in a connected graph [Formula: see text] is called a minimal restrained Steiner set if no proper subset of [Formula: see text] is a restrained Steiner set of [Formula: see text]. The upper restrained Steiner number [Formula: see text] is the maximum cardinality of a minimal restrained Steiner set of [Formula: see text]. The upper restrained Steiner number of certain classes of graphs are determined. Connected graphs of order [Formula: see text] with upper restrained Steiner number [Formula: see text] or [Formula: see text] are characterized. It is shown that for every pair of integers [Formula: see text] and [Formula: see text], with [Formula: see text], there exists a connected graph [Formula: see text] such that [Formula: see text] and [Formula: see text]. Also, it is shown that for every pair of integers [Formula: see text] and [Formula: see text] with [Formula: see text] there exists a connected graph [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] is the upper restrained geodetic number of the graph [Formula: see text].

The open detour number of a graph

2020 ◽  
pp. 2050088
Author(s):  
J. John ◽  
V. R. Sunil Kumar
Keyword(s):  
Connected Graph ◽  
Internal Vertex ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Geodetic Number ◽  
Number Formula ◽  
Positive Integers

A set [Formula: see text] is called an open detour set of [Formula: see text] if for each vertex [Formula: see text] in [Formula: see text], either (1) [Formula: see text] is a detour simplicial vertex of [Formula: see text] and [Formula: see text] or (2) [Formula: see text] is an internal vertex of an [Formula: see text]-[Formula: see text] detour for some [Formula: see text]. An open detour set of minimum cardinality is called a minimum open detour set and this cardinality is the open detour number of [Formula: see text], denoted by [Formula: see text]. Connected graphs of order [Formula: see text] with open detour number [Formula: see text] or [Formula: see text] are characterized. It is shown that for any two positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], there exists a connected graph [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] is the detour number of [Formula: see text]. It is also shown that for every pair of positive integers [Formula: see text] and [Formula: see text] with [Formula: see text] and [Formula: see text], there exists a connected graph [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] is the open geodetic number of [Formula: see text].

The forcing near geodetic number of a graph

2020 ◽  
Vol 12 (02) ◽  
pp. 2050029
Author(s):  
R. Lenin
Keyword(s):  
Connected Graph ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Unique Minimum ◽  
Geodetic Number ◽  
Forcing Number ◽  
Geodetic Set ◽  
Number Formula ◽  
Positive Integers

A set [Formula: see text] is a near geodetic set if for every [Formula: see text] in [Formula: see text] there exist some [Formula: see text] in [Formula: see text] with [Formula: see text] The near geodetic number [Formula: see text] is the minimum cardinality of a near geodetic set in [Formula: see text] A subset [Formula: see text] of a minimum near geodetic set [Formula: see text] is called the forcing subset of [Formula: see text] if [Formula: see text] is the unique minimum near geodetic set containing [Formula: see text]. The forcing number [Formula: see text] of [Formula: see text] in [Formula: see text] is the minimum cardinality of a forcing subset for [Formula: see text], while the forcing near geodetic number [Formula: see text] of [Formula: see text] is the smallest forcing number among all minimum near geodetic sets of [Formula: see text]. In this paper, we initiate the study of forcing near geodetic number of connected graphs. We characterize graphs with [Formula: see text]. Further, we compare the parameters geodetic number[Formula: see text] near geodetic number[Formula: see text] forcing near geodetic number and we proved that, for every positive integers [Formula: see text] and [Formula: see text] with [Formula: see text] and [Formula: see text], there exists a nontrivial connected graph [Formula: see text] with [Formula: see text] [Formula: see text] and [Formula: see text].

scholarly journals The upper connected edge geodetic number of a graph

Filomat ◽  
2012 ◽  
Vol 26 (1) ◽  
pp. 131-141 ◽  
Author(s):  
A.P. Santhakumaran ◽  
J. John
Keyword(s):  
Connected Graph ◽  
Proper Subset ◽  
Maximum Cardinality ◽  
Minimum Order ◽  
Minimum Cardinality ◽  
Geodetic Number ◽  
Geodetic Set ◽  
Positive Integers

For a non-trivial connected graph 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 and any edge geodetic set of order g1(G) is an edge geodetic basis. A connected edge geodetic set of G is an edge geodetic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected edge geodetic set of G is the connected edge geodetic number of G and is denoted by g1c(G). A connected edge geodetic set of cardinality g1c(G) is called a g1c- set of G or connected edge geodetic basis of G. A connected edge geodetic set S in a connected graph G is called a minimal connected edge geodetic set if no proper subset of S is a connected edge geodetic set of G. The upper connected edge geodetic number g+ 1c(G) is the maximum cardinality of a minimal connected edge geodetic set of G. Graphs G of order p for which g1c(G) = g+1c = p are characterized. For positive integers r,d and n ( d + 1 with r ? d ? 2r, there exists a connected graph of radius r, diameter d and upper connected edge geodetic number n. It is shown for any positive integers 2 ? a < b ? c, there exists a connected graph G such that g1(G) = a; g1c(G) = b and g+ 1c(G) = c.

scholarly journals The edge geodetic self decomposition number of a graph

2020 ◽  
Author(s):  
John J ◽  
Stalin D
Keyword(s):  
Connected Graph ◽  
Maximum Cardinality ◽  
Connected Graphs ◽  
Edge Disjoint ◽  
General Properties ◽  
Geodetic Number ◽  
Simple Connected Graph ◽  
Decomposition Number

Let  G = (V, E)  be a simple connected  graph  of order  p and  size q.  A decomposition  of a graph  G is a collection  π  of edge-disjoint sub graphs  G1, G2, ..., Gn  of G such  that every  edge of G belongs to exactly  one Gi , (1 ≤ i ≤ n) . The decomposition  π = {G1, G2, ....Gn } of a connected  graph  G is said to be an edge geodetic self decomposi- tion  if ge (Gi ) = ge (G), (1 ≤ i ≤ n).The maximum  cardinality of π is called the edge geodetic self decomposition  number of G and is denoted by πsge (G), where ge (G) is the edge geodetic number  of G.  Some general properties   satisfied  by  this  concept  are  studied.    Connected  graphs which are edge geodetic self decomposable  are characterized.

Upper triangle free detour number of a graph

2021 ◽  
pp. 2150094
Author(s):  
S. Sethu Ramalingam ◽  
S. Athisayanathan
Keyword(s):  
Free Path ◽  
Connected Graph ◽  
Proper Subset ◽  
Maximum Order ◽  
Minimum Order ◽  
Geodetic Number ◽  
Number Formula ◽  
Positive Integers ◽  
Distance Formula

For any two vertices [Formula: see text] and [Formula: see text] in a connected graph [Formula: see text], the [Formula: see text] path [Formula: see text] is called a [Formula: see text] triangle free path if no three vertices of [Formula: see text] induce a triangle. The triangle free detour distance [Formula: see text] is the length of a longest [Formula: see text] triangle free path in [Formula: see text]. A [Formula: see text] path of length [Formula: see text] is called a [Formula: see text] triangle free detour. A set [Formula: see text] is called a triangle free detour set of [Formula: see text] if every vertex of [Formula: see text] lies on a [Formula: see text] triangle free detour joining a pair of vertices of [Formula: see text]. The triangle free detour number [Formula: see text] of [Formula: see text] is the minimum order of its triangle free detour sets and any triangle free detour set of order [Formula: see text] is a triangle free detour basis of [Formula: see text]. A triangle free detour set [Formula: see text] of [Formula: see text] is called a minimal triangle free detour set if no proper subset of [Formula: see text] is a triangle free detour set of [Formula: see text]. The upper triangle free detour number [Formula: see text] of [Formula: see text] is the maximum order of its minimal triangle free detour sets and any minimal triangle free detour set of order [Formula: see text] is an upper triangle free detour basis of [Formula: see text]. We determine bounds for it and characterize graphs which realize these bounds. For any connected graph [Formula: see text] of order [Formula: see text], [Formula: see text]. Also, for any four positive integers [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] with [Formula: see text], it is shown that there exists a connected graph [Formula: see text] such that [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] is the upper detour number, [Formula: see text] is the upper detour monophonic number and [Formula: see text] is the upper geodetic number of a graph [Formula: see text].

scholarly journals The upper connected vertex detour number of a graph

Filomat ◽  
2012 ◽  
Vol 26 (2) ◽  
pp. 379-388 ◽  
Author(s):  
A.P. Santhakumaran ◽  
P. Titus
Keyword(s):  
Connected Graph ◽  
Proper Subset ◽  
Maximum Cardinality ◽  
Minimum Cardinality ◽  
Positive Integers ◽  
Connected Vertex ◽  
Special Classes

For vertices x and y in a connected graph G = (V, E) of order at least two, the detour distance D(x, y) is the length of the longest x ? y path in G: An x ? y path of length D(x, y) is called an x ? y detour. For any vertex x in G, a set S ? V is an x-detour set of G if each vertex v ? V lies on an x ? y detour for some element y in S: The minimum cardinality of an x-detour set of G is defined as the x-detour number of G; denoted by dx(G): An x-detour set of cardinality dx(G) is called a dx-set of G: 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). A connected x-detour set of cardinality cdx(G) is called a cdx-set of G. A connected x-detour set Sx is called a minimal connected x-detour set if no proper subset of Sx is a connected x-detour set. The upper connected x-detour number, denoted by cd+ x (G), is defined as the maximum cardinality of a minimal connected x-detour set of G: We determine bounds for cd+ x (G) and find the same for some special classes of graphs. For any three integers a; b and c with 2 ? a < b ? c, there is a connected graph G with dx(G) = a; cdx(G) = b and cd+ x (G) = c for some vertex x in G: It is shown that for positive integers R,D and n ? 3 with R < D ? 2R; there exists a connected graph G with detour radius R; detour diameter D and cd+ x (G) = n for some vertex x in G.

scholarly journals The Upper Complement Connected Monophonic Number of a Graph

2019 ◽  
Vol 8 (10S) ◽  
pp. 297-300
Keyword(s):  
Connected Graph ◽  
Proper Subset ◽  
Maximum Cardinality ◽  
Minimum Cardinality ◽  
General Properties ◽  
Positive Integers

For a connected graph a monophonic set of is said to be a complement connected monophonic set if or the subgraph is connected. The minimum cardinality of a complement connected monophonic set of is the complement connected monophonic number of and is denoted by A complement connected monophonic set in a connected graph is called a minimal complement connected monophonic set if no proper subset of is a complement connected monophonic set of . The upper complement connected monophonic number of is the maximum cardinality of a minimal complement connected monophonic set of . Some general properties under this concept are studied. The upper complement connected monophonic number of some standard graphs are determined. Some of its general properties are studied. It is shown that for any positive integers 2 ≤ a ≤b, there exists a connected graph such that ( ) = a and ( ) =b

THE LINEAR GEODETIC NUMBER OF A GRAPH

2011 ◽  
Vol 03 (03) ◽  
pp. 357-368 ◽  
Author(s):  
A. P. SANTHAKUMARAN ◽  
T. JEBARAJ ◽  
S. V. ULLAS CHANDRAN
Keyword(s):  
Connected Graph ◽  
Ordered Set ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Geodetic Number ◽  
Geodetic Set ◽  
Positive Integers

For a connected graph G of order n, an ordered set S = {u1, u2, …, uk} of vertices in G is a linear geodetic set of G if for each vertex x in G, there exists an index i, 1 ≤ i < k such that x lies on a ui - ui + 1 geodesic on G, and a linear geodetic set of minimum cardinality is the linear geodetic number gl(G). The linear geodetic numbers of certain standard graphs are obtained. It is shown that if G is a graph of order n and diameter d, then gl(G) ≤ n - d + 1 and this bound is sharp. For positive integers r, d and k ≥ 2 with r < d ≤ 2r, there exists a connected graph G with rad G = r, diam G = d and gl(G) = k. Also, for integers n, d and k with 2 ≤ d < n, 2 ≤ k ≤ n - d + 1, there exists a connected graph G of order n, diameter d and gl(G) = k. We characterize connected graphs G of order n with gl(G) = n and gl(G) = n - 1. It is shown that for each pair a, b of integers with 3 ≤ a ≤ b, there is a connected graph G with g(G) = a and gl(G) = b. We also discuss how the linear geodetic number of a graph is affected by adding a pendent edge to the graph.

scholarly journals Further results on the outer connected geodetic number of a graph

2020 ◽  
Vol 108 (122) ◽  
pp. 79-89
Author(s):  
Kathiresan Ganesamoorthy ◽  
Duraisamy Jayanthi
Keyword(s):  
Connected Graph ◽  
Fault Tolerant ◽  
Access Network ◽  
Proper Subset ◽  
Network Communication ◽  
Maximum Cardinality ◽  
Geodetic Number ◽  
Local Access ◽  
Geodetic Set ◽  
Network Fault

For a connected graph G of order at least two, an outer connected geodetic set S in a connected graph G is called a minimal outer connected geodetic set if no proper subset of S is an outer connected geodetic set of G. The upper outer connected geodetic number g+ oc(G) of G is the maximum cardinality of a minimal outer connected geodetic set of G. We determine bounds for it and find the upper outer connected geodetic number of some standard graphs. Some realization results on the upper outer connected geodetic number of a graph are studied. The proposed method can be extended to the identification of beacon vertices towards the network fault-tolerant in wireless local access network communication. Also, another parameter forcing outer connected geodetic number fog(G) of a graph G is introduced and several interesting results and realization theorem are proved.

New bounds on the independence number of connected graphs

2018 ◽  
Vol 10 (05) ◽  
pp. 1850069
Author(s):  
Nader Jafari Rad ◽  
Elahe Sharifi
Keyword(s):  
Lower Bound ◽  
Connected Graph ◽  
Maximum Degree ◽  
Applied Mathematics ◽  
Independence Number ◽  
Independent Set ◽  
Maximum Cardinality ◽  
Connected Graphs ◽  
Cut Vertex

The independence number of a graph [Formula: see text], denoted by [Formula: see text], is the maximum cardinality of an independent set of vertices in [Formula: see text]. [Henning and Löwenstein An improved lower bound on the independence number of a graph, Discrete Applied Mathematics  179 (2014) 120–128.] proved that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] does not belong to a specific family of graphs, then [Formula: see text]. In this paper, we strengthen the above bound for connected graphs with maximum degree at least three that have a non-cut-vertex of maximum degree. We show that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] has a non-cut-vertex of maximum degree then [Formula: see text], where [Formula: see text] is the maximum degree of the vertices of [Formula: see text]. We also characterize all connected graphs [Formula: see text] of order [Formula: see text] and size [Formula: see text] that have a non-cut-vertex of maximum degree and [Formula: see text].

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