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.

EXTREME STEINER GRAPHS

2012 ◽  
Vol 04 (02) ◽  
pp. 1250029 ◽  
Author(s):  
A. P. SANTHAKUMARAN
Keyword(s):  
Connected Graph ◽  
Steiner Tree ◽  
Minimum Order ◽  
Minimum Cardinality ◽  
Geodetic Number ◽  
Geodetic Set ◽  
Positive Integers

For a connected graph G of order p ≥ 2 and a set W ⊆ V(G), a tree T contained in G is a Steiner tree with respect to W if T is a tree of minimum order with W ⊆ V(T). The set S(W) consists of all vertices in G that lie on some Steiner tree with respect to W. The set W is a Steiner set for G if S(W) = V(G). The Steiner number s(G) of G is the minimum cardinality of its Steiner sets and any Steiner set of cardinality s(G) is a minimum Steiner set of G. A geodetic set of G is a set S of vertices such that every vertex of G is contained in a geodesic joining some pair of vertices of S. The geodetic number g(G) of G is the minimum cardinality of its geodetic sets and any geodetic set of cardinality g(G) is a minimum geodetic set of G. A vertex v is an extreme vertex of a graph G if the subgraph induced by its neighbors is complete. The number of extreme vertices in G is its extreme order ex (G). A graph G is an extreme Steiner graph if s(G) = ex (G), and an extreme geodesic graph if g(G) = ex (G). Extreme Steiner graphs of order p with Steiner number p - 1 are characterized. It is shown that every pair a, b of integers with 0 ≤ a ≤ b is realizable as the extreme order and Steiner number, respectively, of some graph. For positive integers r, d and l ≥ 2 with r < d ≤ 2r, it is shown that there exists an extreme Steiner graph G of radius r, diameter d, and Steiner number l. For integers p, d and k with 2 ≤ d < p, 2 ≤ k < p and p - d - k + 2 ≥ 0, there exists an extreme Steiner graph G of order p, diameter d and Steiner number k. It is shown that for every pair a, b of integers with 3 ≤ a < b and b = a + 1, there exists an extreme Steiner graph G with s(G) = a and g(G) = b that is not an extreme geodesic graph. It is shown that for every pair a, b of integers with 3 ≤ a < b, there exists an extreme geodesic graph G with g(G) = a and s(G) = b that is not an extreme Steiner graph.

scholarly journals On the upper geodetic global domination number of a graph

2020 ◽  
Vol 39 (6) ◽  
pp. 1627-1647
Author(s):  
X. Lenin Xaviour ◽  
S. Robinson Chellathurai
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Proper Subset ◽  
Maximum Cardinality ◽  
General Properties ◽  
Geodetic Set ◽  
Positive Integers ◽  
Global Domination Number

A set S of vertices in a connected graph G = (V, E) is called a geodetic set if every vertex not in S lies on a shortest path between two vertices from S. A set D of vertices in G is called a dominating set of G if every vertex not in D has at least one neighbor in D. A set D is called a global dominating set in G if S is a dominating set of both G and Ḡ. A set S is called a geodetic global dominating set of G if S is both geodetic and global dominating set of G. A geodetic global dominating set S in G is called a minimal geodetic global dominating set if no proper subset of S is itself a geodetic global dominating set in G. The maximum cardinality of a minimal geodetic global dominating set in G is the upper geodetic global domination number Ῡg+(G) of G. In this paper, the upper geodetic global domination number of certain connected graphs are determined and some of the general properties are studied. It is proved that for all positive integers a, b, p where 3 ≤ a ≤ b < p, there exists a connected graph G such that Ῡg(G) = a, Ῡg+(G) = b and |V (G)| = p.

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 edge-to-edge geodetic domination number of a graph

2021 ◽  
Vol 40 (3) ◽  
pp. 635-658
Author(s):  
J. John ◽  
V. Sujin Flower
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
General Properties ◽  
Geodetic Number ◽  
Geodetic Set ◽  
Positive Integers

Let G = (V, E) be a connected graph with at least three vertices. A set S ⊆ E(G) is called an edge-to-edge geodetic dominating set of G if S is both an edge-to-edge geodetic set of G and an edge dominating set of G. The edge-to-edge geodetic domination number γgee(G) of G is the minimum cardinality of its edge-to-edge geodetic dominating sets. Some general properties satisfied by this concept are studied. Connected graphs of size m with edge-to-edge geodetic domination number 2 or m or m − 1 are characterized. We proved that if G is a connected graph of size m ≥ 4 and Ḡ is also connected, then 4 ≤ γgee(G) + γgee(Ḡ) ≤ 2m − 2. Moreover we characterized graphs for which the lower and the upper bounds are sharp. It is shown that, for every pair of positive integers a, b with 2 ≤ a ≤ b, there exists a connected graph G with gee(G) = a and γgee(G) = b. Also it is shown that, for every pair of positive integers a and b with 2 < a ≤ b, there exists a connected graph G with γe(G) = a and γgee(G) = b, where γe(G) is the edge domination number of G and gee(G) is the edge-to-edge geodetic number of G.

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 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 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 Extreme Monophonic Graphs and Extreme Geodesic Graphs

2016 ◽  
Vol 47 (4) ◽  
pp. 393-404
Author(s):  
P. Titus ◽  
A.P Santhakumaran
Keyword(s):  
Connected Graph ◽  
Minimum Cardinality ◽  
Characterization Result ◽  
Geodetic Number ◽  
Geodetic Set ◽  
Positive Integers ◽  
Chordless Path

For a connected graph $G=(V,E)$ of order at least two, a chord of a path $P$ is an edge joining two non-adjacent vertices of $P$. A path $P$ is called a monophonic path if it is a chordless path. A monophonic set of $G$ is a set $S$ of vertices such that every vertex of $G$ lies on a monophonic path joining some pair of vertices in $S$. The monophonic number of $G$ is the minimum cardinality of its monophonic sets and is denoted by $m(G)$. A geodetic set of $G$ is a set $S$ of vertices such that every vertex of $G$ lies on a geodesic joining some pair of vertices in $S$. The geodetic number of $G$ is the minimum cardinality of its geodetic sets and is denoted by $g(G)$. The number of extreme vertices in $G$ is its extreme order $ex(G)$. A graph $G$ is an extreme monophonic graph if $m(G)=ex(G)$ and an extreme geodesic graph if $g(G)=ex(G)$. Extreme monophonic graphs of order $p$ with monophonic number $p$ and $p-1$ are characterized. It is shown that every pair $a,b$ of integers with $0 \leq a \leq b$ is realized as the extreme order and monophonic number, respectively, of some graph. For positive integers $r,d$ and $k \geq 3$ with $r < d$, it is shown that there exists an extreme monophonic graph $G$ of monophonic radius $r$, monophonic diameter $d$, and monophonic number $k$. Also, we give a characterization result for a graph $G$ which is both extreme geodesic and extreme monophonic.

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