A graph G is called a PMNL-graph if it has a perfect minimum-neighborhood labeling. Let denote the Cartesian product of graphs and . It is also proved in [1] that the Cartesian product of two forests has game chromatic number at most 12 and the Cartesian product of two planar graphs has game chromatic number at most 650. In this paper, we modify the approach of Clark and Suen to prove a variety of similar bounds related to total and paired domination, and also extend these bounds to the n-Cartesian product of graphs A1 through An. A set of vertices Sof a connected graph Gis a nonseparating independent set if Sis independent and G Sis connected. A list of all graphs and graph structures (other than isomorphism class representatives) in this database is available via tab completion. The curves and questions AMS classifications: 05C76, 16Y60, 05C25 . It is also proved in [1] that the Cartesian product of two forests has game chromatic number at most 12 and the Cartesian product of two planar graphs has game chromatic number at most 650. j) (u. i+1,v. graphs of equations given in Cartesian form, polar form, or parametrically. The cartesian product of $$2$$ non-empty sets $$A$$ and $$B$$ is the set of all possible ordered pairs where the first component is from $$A$$ and the second component is from $$B.$$ According to Imrich and Klavzar [4] Cartesian products of graphs were defined in 1912 by Whitehead and Russell [5]. Definition 6 (see [ 4, 13 18 ]). Moreover, such a factorization is unique up to reordering of the factors. is paper gives a detailed study of Cartesian product and factorization of circulant graphs similar to the theory of product and factorization of natural numbers. The Cartesian product of G and H is a graph, which we denote by G 2 H, such that: (i) V.G 2 H/VDV.G/ V.H/, the Cartesian product of the sets V.G/ and V.H/; and (ii) f.g1;h1/;.g2;h2/g2E.G 2 H/if and only if either: (a) g1 Dg2 and fh1;h2g2E.H/; or (b) h1 Dh2 and fg1;g2g2E.G/. Cartesian products of graphs. The Cartesian product of two graphs Gand H, denoted by G H, is a graph with vertex set V(G) V(H), and (a;x)(b;y) 2E(G H) if either ab2E(G) and Research is partially supported by the Iran National Science Foundation (INSF). From specialists in the field, you will learn about interesting connections and recent developments in the field of graph theory by looking in particular at Cartesian products-arguably the most important of the four standard graph products. a given graph is a subgraph (or induced subgraph) of a hypercube, which is the simplest Cartesian product graph. The operation is associative and commutative. A graph G is prime with respect to if G cannot The restriction of the Cartesian product to graphs coincides with the usual Cartesian graph product. In [7] it was proved that (G) (G)2 holds for any graph with (G) 3 and dierent from K4 and K5. Graph of the function Graph of the function over the interval [2,+3]. A sample of this research is [2,3,9,13]. (There is an analogous result for m-regular cartesian products of regular bipartite graphs, but we leave discussion of that to Section 2.) graphs. The paired-domination number pr (G) of G is the minimum cardinality of a paired-dominating set.
Do you navigate arXiv using a screen reader or other assistive technology? This result was generalized by Wilfried Imrich [9] Thus, the Cartesian product of two hypercube graphs is another hypercube: Q i Q j = Q i+j. Cartesian Product is the multiplication of two sets to form the set of all ordered pairs. The first element of the ordered pair belong to first set and second pair belong the second set. For an example, Here, set A and B is multiplied to get Cartesian product AB. The first element of AB is a ordered pair (dog, meat) where dog belongs to set A. This generalizes result on G x K2 obtatned by Chartrand and in 1 Introduction Let G be connected simple graph u. t' V'(G). Introduced by Sabidussi [18] in 1959, it has been applied in many areas since then, for example in space structures [14] and interconnection networks [5]. The applications of Cartesian product can be found in coding theory. For more details on circulant graphs, see [ , ]. An independent transversal dominating set of a graph G is a set $$S \subset V(G)$$ that both dominates G and intersects every maximum independent set of G, and $$\gamma _\mathrm{{it}}(G)$$ is defined to be the minimum cardinality of an independent transversal dominating set of G.In this paper, we investigate how local changes to a graph effect the Cartesian product of two graphs.