Complementary Prism Research Articles

K-Independence on complementary prism graphs

k-Independence on complementary prism graphs

Hardness and efficiency on [formula omitted]-admissibility for graph operations

The t-admissibility problem aims to decide whether a graph G has a spanning tree T in which the distance between any two adjacent vertices of G is at most t. In this case, G is called t-admissible, and the smallest t for which G is t-admissible is the stretch index of G. A complementary prism of G, denoted by GG¯, is obtained by the union of G with its complement G¯ and the addition of a perfect matching between corresponding vertices of G and G¯. One of the challenges of the t-admissibility problem is to determine 3-admissible graph classes, since the computational complexity of such a problem remains open for more than 25 years. Moreover, it is known that recognizing 4-admissible graphs is, in general, an NP-complete problem (Cai and Corneil, 1995), as well as recognizing t-admissible graphs for graphs with diameter at most t+1, for t≥4 (Papoutsakis, 2013). We prove that any graph G, non-complete graph, can be transformed into a 4-admissible one, by obtaining GG¯. Furthermore, we prove that the stretch indexes of GG¯ graphs are equal to 4, and since they have diameter at most t+1, we present a class for which t-admissibility is solved in polynomial time. GG graphs are the Cartesian product G×K2, defined by the union of two copies of G and the addition of a perfect matching between corresponding vertices of the two graphs G. Interestingly, we prove that for GG graphs, whose definition is very similar to GG¯’s, t-admissibility is NP-complete. Generalizing these constructions, we prove that determining t-admissibility is NP-complete for graphs that have perfect matching.

The geodetic hop domination number of complementary prisms

Let [Formula: see text] be a graph and [Formula: see text] be the complement of [Formula: see text]. The complementary prism [Formula: see text] of [Formula: see text] is the graph formed from the disjoint union of [Formula: see text] and [Formula: see text] by adding the edges of a perfect matching between the corresponding vertices of [Formula: see text] and [Formula: see text]. A subset [Formula: see text] of vertices in a connected graph [Formula: see text] is called a geodetic hop dominating set of [Formula: see text] if [Formula: see text] is both a geodetic set and a hop dominating set of [Formula: see text]. The minimum cardinality of a geodetic hop dominating set of [Formula: see text] is its geodetic hop domination number and is denoted by [Formula: see text]. In this paper, we study the geodetic hop domination number of complementary prisms.

On the General Position Number of Complementary Prisms

The general position number gp(G) of a graph G is the cardinality of a largest set of vertices S such that no element of S lies on a geodesic between two other elements of S. The complementary prism G G ¯ of G is the graph formed from the disjoint union of G and its complement G ¯ by adding the edges of a perfect matching between them. It is proved that gp(G G ¯ ) ≤ n(G) + 1 if G is connected and gp(G G ¯ ) ≤ n(G) if G is disconnected. Graphs G for which gp(G G ¯ ) = n(G) + 1 holds, provided that both G and G ¯ are connected, are characterized. A sharp lower bound on gp(G G ¯ ) is proved. If G is a connected bipartite graph or a split graph then gp(G G ¯ ) ∈ {n(G), n(G)+1}. Connected bipartite graphs and block graphs for which gp(G G ¯ ) = n(G) + 1 holds are characterized. A family of block graphs is constructed in which the gp-number of their complementary prisms is arbitrary smaller than their order.

Remarks on k-Clique, k-Independent Set and 2-Contamination in Complementary Prisms

Complementary prism graphs arise from the disjoint union of a graph [Formula: see text] and its complement [Formula: see text] by adding the edges of a perfect matching joining pairs of corresponding vertices of [Formula: see text] and [Formula: see text]. Classical graph problems such as Clique and Independent Set were proved to be NP-complete on such a class of graphs. In this work, we study the complexity of both problems on complementary prism graphs from the parameterized complexity point of view. First, we prove that both problems admit a kernel and therefore are fixed-parameter tractable (FPT) when parameterized by the size of the solution, [Formula: see text]. Then, we show that [Formula: see text]-Clique and [Formula: see text]-Independent Set on complementary prisms do not admit polynomial kernel when parameterized by [Formula: see text], unless [Formula: see text]. Furthermore, we address the [Formula: see text]-Contamination problem in the context of complementary prisms. This problem consists in completely contaminating a given graph [Formula: see text] using a minimum set of initially infected vertices. For a vertex to be contaminated, it is enough that at least two of its neighbors are contaminated. The propagation of the contamination follows this rule until no more vertex can be contaminated. It is known that the minimum set of initially contaminated vertices necessary to contaminate a complementary prism of connected graphs [Formula: see text] and [Formula: see text] has cardinality at most [Formula: see text]. In this paper, we show that the tight upper bound for this invariant on complementary prisms is [Formula: see text], improving a result of Duarte et al. (2017).

On the geodetic hull number for complementary prisms II

In the geodetic convexity, a set of vertices S of a graph G is convex if all vertices belonging to any shortest path between two vertices of S lie in S. The convex hull H(S) of S is the smallest convex set containing S. If H(S) = V (G), then S is a hull set. The cardinality h(G) of a minimum hull set of G is the hull number of G. The complementary prism GḠ of a graph G arises from the disjoint union of the graph G and Ḡ by adding the edges of a perfect matching between the corresponding vertices of G and Ḡ. A graph G is autoconnected if both G and Ḡ are connected. Motivated by previous work, we study the hull number for complementary prisms of autoconnected graphs. When G is a split graph, we present lower and upper bounds showing that the hull number is unlimited. In the other case, when G is a non-split graph, it is limited by 3.

Differential in complementary prisms

AbstractLet G = (V, E) be a graph of order n and let B(D) be the set of vertices in V\D that have a neighbor in the set D. The differential of a set D is defined as ∂(D) = |B(D)| − |D| and the differential of a graph to equal the maximum value of ∂(D) for any subset D of V. A set D of vertices of a graph G is said to be a dominating set if every vertex in V\D is adjacent to a vertex in D. G is a dominant differential graph if it contains a ∂‐set which is also a dominating set. Let be the complement of a graph G. The complementary prism of G is the graph formed from the disjoint union of G and by adding the edges of a perfect matching between the corresponding vertices of G and . This paper is devoted to the computation of differential of complementary prisms . Particular attention is given to the complementary prims of special types of graphs and dominant differential complementary prisms are recognized. Furthermore, a sharp lower bound on the differential of the complementary prism of a graph G in terms of the order of G is presented and the graphs attaining this lower bound are characterized. Finally, the graphs G are characterized for which is small.

Independent strong domination in complementary prisms

Let G = ( V, E ) be a graph and u,v ∈ V . Then, u strongly dominates v if (i) uv ∈ E and (ii) deg( u ) ≥ deg( v ). A set D ⊂ V is a strong-dominating set of G if every vertex in V - D is strongly dominated by at least one vertex in D . A set D ⊆ V is an independent set if no two vertices of D are adjacent. The independent strong domination number i s ( G ) of a graph G is the minimum cardinality of a strong dominating set which is independent. Let Ġ be the complement of a graph G . The complementary prism GĠ of G is the graph formed from the disjoint union of G and Ġ by adding the edges of a perfect matching between the corresponding vertices of G and Ġ . In this paper, we consider the independent strong domination in complementary prisms, characterize the complementary prisms with small independent strong domination numbers, and investigate the relationship between independent strong domination number and the distance-based parameters.

Injective coloring of complementary prism and generalized complementary prism graphs

The complementary prism [Formula: see text] of a graph [Formula: see text] is the graph obtained by drawing edges between the corresponding vertices of a graph [Formula: see text] and its complement [Formula: see text]. In this paper, we generalize the concept of complementary prisms of graphs and determine the injective chromatic number of generalized complementary prisms of graphs. We prove that for any simple graph [Formula: see text] of order [Formula: see text], [Formula: see text] and if [Formula: see text] is a graph with a universal vertex, then [Formula: see text].

Betweenness Centrality of Some Complementary Prism Graphs

There are a lot of centrality measures that have been introduced for networks. One of them is betweenness centrality. It is a measure of the influence of a vertex over the flow of information between all pairs of vertices. This information flows over the shortest paths between these vertices. The fact that any vertex has a high value of centrality indicates that what level this vertex is in connection with vertices which are not adjacent with each other. Since this vertex controls flows of information, it has a potential role in the network. In this paper, we study on the betweenness centrality of some complementary prism graphs.

Open-independent, Open-locating-dominating Sets in Complementary Prism Graphs

For a finite, simple, undirected graph G = (V (G),E(G)), an open-dominating set S ⊆ V (G) is such that every vertex in G has at least one neighbor in S. An open-independent, open-locating-dominating set S ⊆ V (G) (OLDOIND-set for short) is such that no two vertices in G have the same set of neighbors in S and each vertex in S is open-dominated exactly once by S. The problem of deciding whether or not a given graph has an OLDOIND-set is known to be NP-complete. The complementary prism of G is the graph GG‾, formed from the disjoint union of G and its complement G‾ by adding the edges of a perfect matching between the corresponding vertices of G and G‾. We provided a logarithmic lower bound on the size of an OLDOIND-set in any graph. Various properties of and bounds on OLDOIND-sets in complementary prisms were presented and the cases of cliques, paths and cycles have been completely solved. It has been shown that for any graph with girth at least five, it can be decided in polynomial time whether or not its complementary prism has an OLD-OIND-set (and also the set can be found in polynomial time if it exists).

Identifying Codes in the Complementary Prism of Cycles

We show that an identifying code of minimum order in the complementary prism of a cycle of order n has order 7n/9 + Θ(1). Furthermore, we observe that the clique-width of the complementary prism of a graph of clique-width k is at most 4k, and discuss some algorithmic consequences.

Some results on the b-chromatic number in complementary prism graphs

A b-coloring of a graph G is a proper coloring of G with k colors such that each color class has a vertex that is adjacent to at least one vertex of every other color classes. The b-chromatic number is the largest integer k for which G has a b-coloring with k colors. In this paper, we present some results on b-coloring in complementary prism graphs.

On the geodetic number of complementary prisms

Motivated by previous work, we prove that, given a graph G and a positive integer k, deciding whether the geodetic number of the complementary prism of G is at most k is an NP-complete problem. We also show lower and upper bounds for the geodetic number of a complementary prism, and determine the geodetic number of a complementary prism when it is defined over a split graph.

Strong Independent Saturation in Complementary Prisms

Strong Independent Saturation in Complementary Prisms

Spectra, signless Laplacian and Laplacian spectra of complementary prisms of graphs

The complementary prism GG‾ of a graph G is obtained from the disjoint union of G and its complement G‾ by adding an edge for each pair of vertices (v,v′), where v is in G and its copy v′ is in G‾. The Petersen graph C5C5‾ and, for n≥2, the corona product of Kn and K1 which is KnKn‾ are examples of complementary prisms. This paper is devoted to the computation of eigenpairs of the adjacency, signless Laplacian and Laplacian matrices of a complementary prism GG‾ in terms of the eigenpairs of the corresponding matrices of G. Particular attention is given to the complementary prisms of regular graphs. Furthermore, Petersen graph is shown to be the unique complementary prism which is a strongly regular graph.

Complexity properties of complementary prisms

The complementary prism $$G\bar{G}$$GG? of a graph G arises from the disjoint union of the graph G and its complement $$\bar{G}$$G? by adding the edges of a perfect matching joining pairs of corresponding vertices of G and $$\bar{G}$$G?. Haynes, Henning, Slater, and van der Merwe introduced the complementary prism and as a variation of the well-known prism. We study algorithmic/complexity properties of complementary prisms with respect to cliques, independent sets, k-domination, and especially $$P_3$$P3-convexity. We establish hardness results and identify some efficiently solvable cases.

Cycles in complementary prisms

The complementary prism GG¯ of a graph G arises from the disjoint union of G and the complement G¯ of G by adding a perfect matching joining corresponding pairs of vertices in G and G¯. Partially answering a question posed by Haynes et al. (2007) we provide an efficient characterization of the circumference of the complementary prism TT¯ of a tree T and show that TT¯ has cycles of all lengths between 3 and its circumference. Furthermore, we prove that for a given graph of bounded maximum degree it can be decided in polynomial time whether its complementary prism is Hamiltonian.

On the Average Lower Domination Number and Some Results of Complementary Prisms and Graph Join

On the Average Lower Domination Number and Some Results of Complementary Prisms and Graph Join

Managing the Innovation Supply Chain to Maximize Personalized Medicine

Personalized medicine epitomizes an evolving model of care tailored to the individual patient. This emerging paradigm harnesses radical technological advances to define each patient's molecular characteristics and decipher his or her unique pathophysiological processes. Translated into individualized algorithms, personalized medicine aims to predict, prevent, and cure disease without producing therapeutic adverse events. Although the transformative power of personalized medicine is generally recognized by physicians, patients, and payers, the complexity of translating discoveries into new modalities that transform health care is less appreciated. We often consider the flow of innovation and technology along a continuum of discovery, development, regulation, and application bridging the bench with the bedside. However, this process also can be viewed through a complementary prism, as a necessary supply chain of services and providers, each making essential contributions to the development of the final product to maximize value to consumers. Considering personalized medicine in this context of supply chain management highlights essential points of vulnerability and/or scalability that can ultimately constrain translation of the biological revolution or potentiate it into individualized diagnostics and therapeutics for optimized value creation and delivery.