scholarly journals Weakly convex domination subdivision number of a graph

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2101-2110
Author(s):  
Magda Dettlaff ◽  
Saeed Kosary ◽  
Magdalena Lemańska ◽  
Seyed Sheikholeslami
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Weakly Convex ◽  
Minimum Cardinality ◽  
Minimum Number ◽  
Convex Domination Number ◽  
Domination Subdivision Number

A set X is weakly convex in G if for any two vertices a,b ? X there exists an ab-geodesic such that all of its vertices belong to X. A set X ? V is a weakly convex dominating set if X is weakly convex and dominating. The weakly convex domination number ?wcon(G) of a graph G equals the minimum cardinality of a weakly convex dominating set in G. The weakly convex domination subdivision number sd?wcon (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the weakly convex domination number. In this paper we initiate the study of weakly convex domination subdivision number and establish upper bounds for it.

scholarly journals Independent Domination Subdivision in Graphs

2021 ◽  
Author(s):  
Ammar Babikir ◽  
Magda Dettlaff ◽  
Michael A. Henning ◽  
Magdalena Lemańska
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Independent Set ◽  
Minimum Cardinality ◽  
Minimum Number ◽  
Independent Domination ◽  
Delta G ◽  
Independent Dominating Set ◽  
Domination Subdivision Number

AbstractA set S of vertices in a graph G is a dominating set if every vertex not in S is ad jacent to a vertex in S. If, in addition, S is an independent set, then S is an independent dominating set. The independent domination number i(G) of G is the minimum cardinality of an independent dominating set in G. The independent domination subdivision number $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( G ) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the independent domination number. We show that for every connected graph G on at least three vertices, the parameter $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( G ) is well defined and differs significantly from the well-studied domination subdivision number $$\mathrm{sd_\gamma }(G)$$ sd γ ( G ) . For example, if G is a block graph, then $$\mathrm{sd_\gamma }(G) \le 3$$ sd γ ( G ) ≤ 3 , while $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( G ) can be arbitrary large. Further we show that there exist connected graph G with arbitrarily large maximum degree $$\Delta (G)$$ Δ ( G ) such that $$ \hbox {sd}_{\mathrm{i}}(G) \ge 3 \Delta (G) - 2$$ sd i ( G ) ≥ 3 Δ ( G ) - 2 , in contrast to the known result that $$\mathrm{sd_\gamma }(G) \le 2 \Delta (G) - 1$$ sd γ ( G ) ≤ 2 Δ ( G ) - 1 always holds. Among other results, we present a simple characterization of trees T with $$ \hbox {sd}_{\mathrm{i}}(T) = 1$$ sd i ( T ) = 1 .

scholarly journals Locating Equitable Domination and Independence Subdivision Numbers of Graphs

2014 ◽  
Vol 9 ◽  
pp. 27-32 ◽  
Author(s):  
P. Sumathi ◽  
G. Alarmelumangai
Keyword(s):  
Dominating Set ◽  
Independence Number ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Minimum Number ◽  
Domination Subdivision Number

Let G = (V,E) be a simple, undirected, finite nontrivial graph. A non empty set DV of vertices in a graph G is a dominating set if every vertex in V-D is adjacent to some vertex in D. The domination number (G) is the minimum cardinality of a dominating set of G. A dominating set D is a locating equitable dominating set of G if for any two vertices u,wєV-D, N(u)∩D ≠ N(w)∩D, |N(u)∩D| ≠ |N(w)∩D|. The locating equitable domination number of G is the minimum cardinality of a locating equitable dominating set of G. The locating equitable domination subdivision number of G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the locating equitable domination number and is denoted by sdγle(G). The independence subdivision number sdβle(G) to equal the minimum number of edges that must be subdivided in order to increase the independence number. In this paper, we establish bounds on sdγle(G) and sdβle(G) for some families of graphs.

Secure domination subdivision number of a graph

2019 ◽  
Vol 11 (03) ◽  
pp. 1950036
Author(s):  
S. V. Divya Rashmi ◽  
A. Somasundaram ◽  
S. Arumugam
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Minimum Number ◽  
Number Formula ◽  
Secure Domination ◽  
Domination Subdivision Number

Let [Formula: see text] be a graph of order [Formula: see text] and size [Formula: see text] A dominating set [Formula: see text] of [Formula: see text] is called a secure dominating set if for each [Formula: see text] there exists [Formula: see text] such that [Formula: see text] is adjacent to [Formula: see text] and [Formula: see text] is a dominating set of [Formula: see text] In this case, we say that [Formula: see text] is [Formula: see text]-defended by [Formula: see text] or [Formula: see text] [Formula: see text]-defends [Formula: see text] The secure domination number [Formula: see text] is the minimum cardinality of a secure dominating set of [Formula: see text] The secure domination subdivision number of [Formula: see text] is the minimum number of edges that must be subdivided (each edge in [Formula: see text] can be subdivided at most once) in order to increase the secure domination number. In this paper, we present several results on this parameter.

scholarly journals On the Paired-Domination Subdivision Number of Trees

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1135
Author(s):  
Shouliu Wei ◽  
Guoliang Hao ◽  
Seyed Mahmoud Sheikholeslami ◽  
Rana Khoeilar ◽  
Hossein Karami
Keyword(s):  
Perfect Matching ◽  
Dominating Set ◽  
Domination Number ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Paired Domination Number ◽  
Minimum Number ◽  
Paired Domination ◽  
Domination Subdivision Number

A paired-dominating set of a graph G without isolated vertices is a dominating set of vertices whose induced subgraph has perfect matching. The minimum cardinality of a paired-dominating set of G is called the paired-domination number γpr(G) of G. The paired-domination subdivision number sdγpr(G) of G is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the paired-domination number. Here, we show that, for each tree T≠P5 of order n≥3 and each edge e∉E(T), sdγpr(T)+sdγpr(T+e)≤n+2.

TOTAL OUTER-CONNECTED DOMINATION SUBDIVISION NUMBERS IN GRAPHS

2013 ◽  
Vol 05 (03) ◽  
pp. 1350009
Author(s):  
O. FAVARON ◽  
R. KHOEILAR ◽  
S. M. SHEIKHOLESLAMI
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Connected Dominating Set ◽  
Minimum Size ◽  
Connected Domination Number ◽  
Minimum Number ◽  
Domination Subdivision Number

A set S of vertices of a graph G is a total outer-connected dominating set if every vertex in V(G) is adjacent to some vertex in S and the subgraph G[V\S] induced by V\S is connected. The total outer-connected domination numberγ toc (G) is the minimum size of such a set. The total outer-connected domination subdivision number sd γ toc (G) is the minimum number of edges that must be subdivided in order to increase the total outer-connected domination number. We prove the existence of sd γ toc (G) for every connected graph G of order at least 3 and give upper bounds on it in some classes of graphs.

Domination Theory in Graphs

2020 ◽  
pp. 1-23
Author(s):  
E. Sampathkumar ◽  
L. Pushpalatha
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Correct Conclusion ◽  
Minimum Number ◽  
Domination In Graphs

The study of domination in graphs originated around 1850 with the problems of placing minimum number of queens or other chess pieces on an n x n chess board so as to cover/dominate every square. The rules of chess specify that in one move a queen can advance any number of squares horizontally, vertically, or diagonally as long as there are no other chess pieces in its way. In 1850 enthusiasts who studied the problem came to the correct conclusion that all the squares in an 8 x 8 chessboard can be dominated by five queens and five is the minimum such number. With very few exceptions (Rooks, Bishops), these problems still remain unsolved today. Let G = (V,E) be a graph. A set S ⊂ V is a dominating set of G if every vertex in V–S is adjacent to some vertex in D. The domination number γ(G) of G is the minimum cardinality of a dominating set.

scholarly journals On restricted domination in graphs

2007 ◽  
Vol 57 (5) ◽  
Author(s):  
Vladimir Samodivkin
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Minimum Number ◽  
Domination In Graphs ◽  
Restricted Domination Number

AbstractThe k-restricted domination number of a graph G is the minimum number d k such that for any subset U of k vertices of G, there is a dominating set in G including U and having at most d k vertices. Some new upper bounds in terms of order and degrees for this number are found.

On the dominator coloring in proper interval graphs and block graphs

2015 ◽  
Vol 07 (04) ◽  
pp. 1550043 ◽  
Author(s):  
B. S. Panda ◽  
Arti Pandey
Keyword(s):  
Chromatic Number ◽  
Dominating Set ◽  
Domination Number ◽  
Maximum Size ◽  
Interval Graphs ◽  
Proper Coloring ◽  
Minimum Cardinality ◽  
Block Graphs ◽  
Minimum Number ◽  
Dominator Coloring

In a graph [Formula: see text], a vertex [Formula: see text] dominates a vertex [Formula: see text] if either [Formula: see text] or [Formula: see text] is adjacent to [Formula: see text]. A subset of vertex set [Formula: see text] that dominates all the vertices of [Formula: see text] is called a dominating set of graph [Formula: see text]. The minimum cardinality of a dominating set of [Formula: see text] is called the domination number of [Formula: see text] and is denoted by [Formula: see text]. A proper coloring of a graph [Formula: see text] is an assignment of colors to the vertices of [Formula: see text] such that any two adjacent vertices get different colors. The minimum number of colors required for a proper coloring of [Formula: see text] is called the chromatic number of [Formula: see text] and is denoted by [Formula: see text]. A dominator coloring of a graph [Formula: see text] is a proper coloring of the vertices of [Formula: see text] such that every vertex dominates all the vertices of at least one color class. The minimum number of colors required for a dominator coloring of [Formula: see text] is called the dominator chromatic number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we study the dominator chromatic number for the proper interval graphs and block graphs. We show that every proper interval graph [Formula: see text] satisfies [Formula: see text], and these bounds are sharp. For a block graph [Formula: see text], where one of the end block is of maximum size, we show that [Formula: see text]. We also characterize the block graphs with an end block of maximum size and attaining the lower bound.

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 New Probabilistic Upper Bounds on the Domination Number of a Graph

10.37236/8345 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Nader Jafari Rad
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Regular Graphs ◽  
Minimum Cardinality

A subset $S$ of vertices of a graph $G$ is a dominating set of $G$ if every vertex in $V(G)-S$ has a neighbor in $S$. The domination number $\gamma(G)$ of $G$ is the minimum cardinality of a dominating set of $G$. In this paper, we obtain new (probabilistic) upper bounds for the domination number of a graph, and improve previous bounds given by Arnautov (1974), Payan (1975), and Caro and Roditty (1985) for any graph, and Harant, Pruchnewski and Voigt (1999) for regular graphs.

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