On the monatic number of a graph (original) (raw)
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On the (M, D) number of a graph
Proyecciones (Antofagasta)
For a connected graph G = (V, E), a monophonic set of G is a set M ⊆ V (G) such that every vertex of G is contained in a monophonic path joining some pair of vertices in M. A subset D of vertices in G is called dominating set if every vertex not in D has at least one neighbour in D. A monophonic dominating set M is both a monophonic and a dominating set. The monophonic,dominating,monophonic domination number m(G), γ(G), γ m (G) respectively are the minimum cardinality of the respective sets in G. Monophonic domination number of certain classes of graphs are determined. Connected graph of order p with monophonic domination number p − 1 or p is characterised. It is shown that for every two intigers a, b ≥ 2 with 2 ≤ a ≤ b, there is a connected graph G such that γ m (G) = a and γ g (G) = b, where γ g (G) is the geodetic domination number of a graph.
Some new results on the b-domatic number of graphs
Electron. J. Graph Theory Appl., 2021
A domatic partition P of a graph G =( V , E ) is a partition of V into classes that are pairwise disjoint dominating sets. Such a partition P is called b -maximal if no larger domatic partition P' can be obtained by gathering subsets of some classes of P to form a new class. The b -domatic number bd ( G ) is the minimum cardinality of a b -maximal domatic partition of G . In this paper, we characterize the graphs G of order n with bd ( G ) ∈ { n -1, n -2, n -3}. Then we prove that for any graph G on n vertices, bd ( G )+ bd ( Ġ ) ≤ n +1, where Ġ is the complement of G . Moreover, we provide a characterization of the graphs G of order n with bd ( G )+ bd ( Ġ ) ∈ { n +1, n } as well as those graphs for which bd ( G )= bd ( Ġ )= n /2.
The Monopoly in the Join of Graphs
Journal of Informatics and Mathematical Sciences, 2018
In a graph \(G = (V,E)\), a set \(M\subseteq V(G)\) is said to be a monopoly set of \(G\) if every vertex \(v\in V-M\) has, at least, \(\frac{d(v)}{2}\) neighbors in \(M\). The monopoly size \(mo(G)\) of \(G\) is the minimum cardinality of a monopoly set among all monopoly sets of \(G\). A join graph is the complete union of two arbitrary graphs. In this paper, we investigate the monopoly set in the join of graphs. As consequences the monopoly size of the join of graphs is obtained. Upper and lower bound of the monopoly size of join graphs are obtained. The exact values of monopoly size for the join of some standard graphs with others are obtained.
Discrete Mathematics, 2000
A graph G is said to have property Pm if it contains no subdivision of Km+1 and no subdivision of K m=2 +1; m=2 +1 . Chartrand et al. (J. Combin Theory 10 (1971) 12-41) (see also Problem 6.3 in Jensen and Toft (Graph Coloring Problems, Wiley, New York, 1995) conjectured that the set of vertices (respectively, edges) of any graph with property P m can be partitioned into m − n + 1 subsets such that each of these subsets induces a graph with property Pn, provided m¿n¿1 (respectively, m¿n¿2). We prove that both conjectures fail when m ¿ cn 2 for some positive constant c. In fact, we prove that under the condition m ¿ cn 2 , there exists a graph G with property Pm such that in every colouring of its vertices or edges with m colours there is a monochromatic subgraph H with Hajà os number h(H ) ¿ n, that is, with a subdivision of Kn+1. In addition, we prove bounds of Nordhaus-Gaddum type for the Hajà os number.
The Independent Monopoly Size of Graphs
In a graph G = (V, E), a set D ⊆ V (G) is said to be a monopoly set of G if every vertex v ∈ V −D has at least d (v) 2 neighbors in D. The monopoly size of G, denoted mo(G), is the minimum cardinality of a monopoly set among all monopoly sets in G. The set D ⊆ V (G) is an independent monopoly set in G if it is both a monopoly set and an independent set in G. The number of vertices in a minimum independent monopoly set in a graph G is the independent monopoly size of G and is denoted by imo(G). In this paper, we study the existence of independent monopoly set in graphs, bounds for imo(G), and some exact values for some standard graphs are obtained. Finally we characterize all graphs of order n with imo(G) = 1, n − 1 and n.
On The Fixatic Number of Graphs
2017
The fixing number of a graph GGG is the smallest cardinality of a set of vertices FsubseteqV(G)F\subseteq V(G)FsubseteqV(G) such that only the trivial automorphism of GGG fixes every vertex in FFF. Let Pi\PiPi === F1,F2,ldots,Fk\{F_1,F_2,\ldots,F_k\}F1,F2,ldots,Fk be an ordered kkk-partition of V(G)V(G)V(G). Then Pi\PiPi is called a {\it fixatic partition} if for all iii; 1leqileqk1\leq i\leq k1leqileqk, FiF_iFi is a fixing set for GGG. The cardinality of a largest fixatic partition is called the {\it fixatic number} of GGG. In this paper, we study the fixatic numbers of graphs. Sharp bounds for the fixatic number of graphs in general and exact values with specified conditions are given. Some realizable results are also given in this paper.
Operations on covering numbers of certain graph classes
International Journal of Advanced Mathematical Sciences, 2016
The bounds on the sum and product of chromatic numbers of a graph and its complement are known as Nordhaus-Gaddum inequalities. In this paper, we study the operations on the Independence numbers of graphs with their complement. We also provide a new characterization of certain graph classes.
Bull. ICA, 2020
If X is any nonempty set on n ≥ 2 elements we define the set graph Gn to be the graph whose vertices are the 2 − 2 proper subsets of X with two vertices adjacent if and only if their underlying sets are disjoint. We discuss some properties of Gn. In particular we find its clique partition number and its product dimension. We also give bounds for its representation number. We use standard graph theory terminology as given in [13]. A family of subsets S1, S2, . . . of a set S gives a graph in a natural way if we use these sets as vertices and let SiSj for i 6= j be an edge if and only if the corresponding subsets have a nonempty intersection. In [12], Marczewski has established the converse, i.e. for any graph G there is a set S, such that a family of its subsets defines G according to the above description. Erdős, Goodman and Posa in [1] have remarked that one may replace the idea of a nonempty intersection with disjointness of the subsets since the same would then imply Marczewski’s...