The binary locating-dominating number of some convex polytopes (original) (raw)

The open-locating-dominating number of some convex polytopes

Filomat, 2018

In this paper we will investigate the problem of finding the open-locating-dominating number for some classes of planar graphs - convex polytopes. We considered Dn, Tn, Bn, Cn, En and Rn classes of convex polytopes known from the literature. The exact values of open-locating-dominating number for Dn and Rn polytopes are presented, along with the upper bounds for Tn, Bn, Cn, and En polytopes.

The convex and weak convex domination number of convex polytopes

Revista de la Unión Matemática Argentina, 2022

This paper is devoted to solving the weakly convex dominating set problem and the convex dominating set problem for some classes of planar graphs-convex polytopes. We consider all classes of convex polytopes known from the literature and present exact values of weakly convex and convex domination number for all classes, namely An, Bn, Cn, Dn, En, Rn, R n , Qn, Sn, S n , Tn, T n and Un. When n is up to 26, the values are confirmed by using the exact method, while for greater values of n theoretical proofs are given. 2020 Mathematics Subject Classification. 05C69, 05C38, 05C40. This research was partially supported by the Serbian Ministry of Science under grant no. 174010. 1 A. Kartelj, D. Matić and V. Filipović, Solving the signed Roman domination and signed total Roman domination problems by exact and heuristic methods (unpublished manuscript).

Determination of the element numbers of the regular polytopes

Geometriae Dedicata, 2012

Let be a set of n-dimensional polytopes. A set of n-dimensional polytopes is said to be an element set for if each polytope in is the union of a finite number of polytopes in identified along (n − 1)-dimensional faces. The element number of the set of polyhedra, denoted by e( ), is the minimum cardinality of the element sets for , where the minimum is taken over all possible element sets ∈ E( ). It is proved in Theorem 1 that the element number of the convex regular 4-dimensional polytopes is 4, and in Theorem 2 that the element numbers of the convex regular n-dimensional polytopes is 3 for n ≥ 5. The results in this paper together with our previous papers determine completely the element numbers of the convex regular n-dimensional polytopes for all n ≥ 2.

On the dominating set polytope

European Journal of Combinatorics, 2008

In this paper, we study the dominating set polytope in the class of graphs that decompose by one-node cutsets where the pieces are cycles. We describe some classes of facets and procedures to construct facets of the polytope in that class of graphs, and establish some structural properties. As a consequence we obtain a complete description of the polytope by a system of inequalities when the graph is a cycle. We also show that the separation problem related to that system can be solved in polynomial time. This yields a polynomial time cutting plane algorithm for the minimum weight dominating set problem in that case. We further discuss the applications for the class of cactus graphs.

A problem in enumerating extreme points, and an efficient algorithm for one class of polytopes

Optimization Letters, 2008

We consider the problem of developing an efficient algorithm for enumerating the extreme points of a convex polytope specified by linear constraints. Murty and Chung [8] introduced the concept of a segment of a polytope, and used it to develop some steps for carrying out the enumeration efficiently until the convex hull of the set of known extreme points becomes a segment. That effort stops with a segment, other steps outlined in [8] for carrying out the enumeration after reaching a segment, or for checking whether the segment is equal to the original polytope, do not constitute an efficient algorithm. Here we describe the central problem in carrying out the enumeration efficiently after reaching a segment. We then discuss two procedures for enumerating extreme points, the mukkadvayam checking procedure, and the nearest point procedure. We divide polytopes into two classes: Class 1 polytopes have at least one extreme point satisfying the property that there is a hyperplane H through that extreme point such that every facet of the polytope incident at that extreme point has relative interior point intersections with both sides of H; Class 2 polytopes have the property that every hyperplane through any any extreme point has at least one facet incident at that extreme point completely contained on one of its sides. We then prove that the procedures developed solve the problem efficiently when the polytope belongs to Class 2. The algorithm may also work when the polytope belongs to Class 1, but at the moment we do not have a proof that all its extreme points will be enumerated by the algorithm.

Computing locating-total domination number in some rotationally symmetric graphs

Science Progress

Let [Formula: see text] be a connected graph. A locating-total dominating set in a graph G is a total dominating set S of a G, for every pair of vertices [Formula: see text], such that [Formula: see text]. The minimum cardinality of a locating-total dominating set is called locating-total domination number and represented as [Formula: see text]. In this paper, locating-total domination number is determined for some cycle-related graphs. Furthermore, some well-known graphs of convex polytopes from the literature are also considered for the locating-total domination number.

Computation of Metric Dimension of Certain Subdivided Convex Polytopes

Journal of Mathematics, 2022

The distance d z 1 , z 2 from vertex z 1 ∈ V G to z 2 ∈ V G is minimum length of z 1 , z 2 -path in a given connected graph G having E(G) and V(G) edges and vertices’/nodes’ sets, respectively. Suppose Z = z 1 , z 2 , z 3 , … , z m ⊆ V G is an order set and c ∈ V G , and the code of c with reference to Z is the m-tuple {d(c, z1), d(c, z2), d(c, z13), …, d(c, zk)}. Then, Z is named as the locating set or resolving set if each node of G has unique code. A locating set of least cardinality is described as a basis set for the graph G , and its cardinal number is referred to as metric dimension symbolized by dim G . Metric dimension of certain subdivided convex polytopes S T n has been computed, and it is concluded that just four vertices are sufficient for unique coding of all nodes belonging to this family of convex polytopes.

On the k-dominating set polytope of web graphs

2010

In this work we present some results on the polyhedral structure of the convex hull of integer points in polyhedra of the form {x ≥ 0 : Mx ≥ k1}, for a 0, 1 matrix M and a positive integer number k. In particular, we consider the k-dominating set problem in a graph. Given a graph G = (V, E), a set D ⊆ V is a k-dominating set if every vertex in V is adjacent to at least k vertices of D. The k-dominating set problem consists in finding a k-dominating set of minimum cardinality. The k-dominating set polytope is the convex hull of the incidence vectors of k-dominating sets in G and it is a natural generalization of the well-known dominating set polytope of a graph. We apply our results for general problems to the k-dominating set polytope of some particular families of web graphs.