Restrained geodetic domination of edge subdivision graph

2021 ◽  
pp. 2250002
Author(s):  
John Joy Mulloor ◽  
V. Sangeetha
Keyword(s):  
Connected Graph ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Geodetic Number ◽  
Subdivision Graph ◽  
Geodetic Set

For a connected graph [Formula: see text], a set [Formula: see text] subset of [Formula: see text] is said to be a geodetic set if all vertices in [Formula: see text] should lie in some [Formula: see text] geodesic for some [Formula: see text]. The minimum cardinality of the geodetic set is the geodetic number. In this paper, the authors discussed the geodetic number, geodetic domination number, and the restrained geodetic domination of the edge subdivision graph.

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.

scholarly journals Convex dominating-geodetic partitions in graphs

Filomat ◽  
2016 ◽  
Vol 30 (11) ◽  
pp. 3075-3082
Author(s):  
Yero González ◽  
Magdalena Lemńska
Keyword(s):  
Present Article ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Geodetic Number ◽  
Geodetic Set ◽  
Convex Domination Number

The distance d(u,v) between two vertices u and v in a connected graph G is the length of a shortest u-v path in G. A u-v path of length d(u,v) is called u-v geodesic. A set X is convex in G if vertices from all a -b geodesics belong to X for every two vertices a,b?X. A set of vertices D is dominating in G if every vertex of V-D has at least one neighbor in D. The convex domination number con(G) of a graph G equals the minimum cardinality of a convex dominating set in G. A set of vertices S of a graph G is a geodetic set of G if every vertex v ? S lies on a x-y geodesic between two vertices x,y of S. The minimum cardinality of a geodetic set of G is the geodetic number of G and it is denoted by g(G). Let D,S be a convex dominating set and a geodetic set in G, respectively. The two sets D and S form a convex dominating-geodetic partition of G if |D| + |S| = |V(G)|. Moreover, a convex dominating-geodetic partition of G is called optimal if D is a ?con(G)-set and S is a g(G)-set. In the present article we study the (optimal) convex dominating-geodetic partitions of graphs.

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].

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.

Geodetic global domination in corona and strong product of graphs

2020 ◽  
Vol 12 (04) ◽  
pp. 2050043
Author(s):  
X. Lenin Xaviour ◽  
S. Robinson Chellathurai
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Strong Product ◽  
Geodetic Set ◽  
Product Of Graphs ◽  
Global Domination Number

A set S of vertices in a connected graph [Formula: see text] is called a geodetic set if every vertex not in [Formula: see text] lies on a shortest path between two vertices from [Formula: see text]. A set [Formula: see text] of vertices in [Formula: see text] is called a dominating set of [Formula: see text] if every vertex not in [Formula: see text] has at least one neighbor in [Formula: see text]. A set [Formula: see text] is called a geodetic global dominating set of [Formula: see text] if [Formula: see text] is both geodetic and global dominating set of [Formula: see text]. The geodetic global domination number is the minimum cardinality of a geodetic global dominating set in [Formula: see text]. In this paper, we determine the geodetic global domination number of the corona and strong products of two graphs.

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.

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 Geodetic Global Domination in the Join of Two Graphs

2020 ◽  
Vol 8 (5) ◽  
pp. 4579-4583
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Dominating Number ◽  
Geodetic Set ◽  
Global Domination Number

A set S of vertices in a connected graph is called a geodetic set if every vertex not in lies on a shortest path between two vertices from . A set of vertices in is called a dominating set of if every vertex not in has at least one neighbor in . A set is called a geodetic global dominating set of if is both geodetic and global dominating set of . The geodetic global dominating number is the minimum cardinality of a geodetic global dominating set in . In this paper we determine the geodetic global domination number of the join of two graphs.

scholarly journals Distance Domination and Distance Irredundance in Graphs

10.37236/953 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Adriana Hansberg ◽  
Dirk Meierling ◽  
Lutz Volkmann
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Distance Domination

A set $D\subseteq V$ of vertices is said to be a (connected) distance $k$-dominating set of $G$ if the distance between each vertex $u\in V-D$ and $D$ is at most $k$ (and $D$ induces a connected graph in $G$). The minimum cardinality of a (connected) distance $k$-dominating set in $G$ is the (connected) distance $k$-domination number of $G$, denoted by $\gamma_k(G)$ ($\gamma_k^c(G)$, respectively). The set $D$ is defined to be a total $k$-dominating set of $G$ if every vertex in $V$ is within distance $k$ from some vertex of $D$ other than itself. The minimum cardinality among all total $k$-dominating sets of $G$ is called the total $k$-domination number of $G$ and is denoted by $\gamma_k^t(G)$. For $x\in X\subseteq V$, if $N^k[x]-N^k[X-x]\neq\emptyset$, the vertex $x$ is said to be $k$-irredundant in $X$. A set $X$ containing only $k$-irredundant vertices is called $k$-irredundant. The $k$-irredundance number of $G$, denoted by $ir_k(G)$, is the minimum cardinality taken over all maximal $k$-irredundant sets of vertices of $G$. In this paper we establish lower bounds for the distance $k$-irredundance number of graphs and trees. More precisely, we prove that ${5k+1\over 2}ir_k(G)\geq \gamma_k^c(G)+2k$ for each connected graph $G$ and $(2k+1)ir_k(T)\geq\gamma_k^c(T)+2k\geq |V|+2k-kn_1(T)$ for each tree $T=(V,E)$ with $n_1(T)$ leaves. A class of examples shows that the latter bound is sharp. The second inequality generalizes a result of Meierling and Volkmann and Cyman, Lemańska and Raczek regarding $\gamma_k$ and the first generalizes a result of Favaron and Kratsch regarding $ir_1$. Furthermore, we shall show that $\gamma_k^c(G)\leq{3k+1\over2}\gamma_k^t(G)-2k$ for each connected graph $G$, thereby generalizing a result of Favaron and Kratsch regarding $k=1$.

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