Convex Domination Research Articles

Convex 2-Domination in Graphs

Let $G$ be a connected graph. A set $S \subseteq V(G)$ is \textit{convex $2$-dominating} if $S$ is both convex and $2$-dominating. The minimum cardinality among all convex $2$-dominating sets in $G$, denoted by $\gamma_{2con}(G)$, is called the \textit{convex $2$-domination number} of $G$. In this paper, we initiate the study of convex $2$- domination in graphs. We show that any two positive integers $a$ and $b$ with $6 \le a \le b$ are, respectively, realizable as the convex domination number and convex $2$-domination number of some connected graph. Furthermore, we characterize the convex $2$-dominating sets in the join, corona, lexicographic product, and Cartesian product of two graphs and determine the corresponding convex $2$-domination number of each of these graphs.

On comparisons of information structures with infinite states

Blackwell's theorem on the comparison of information structures is by now sufficiently well-understood for a finite state space, but important gaps remain in the infinite case. While the equivalence of (i) sufficiency and (ii) more-informativeness is known, we present a comprehensive theory that establishes equivalences between these two orders (in both their original and almost all versions) and three additional prior-dependent criteria on general (Polish) state spaces. We consider (iii) Bayesian preference, (iv) convex dominance, and (v) mean-preserving-spread (dilation) for all priors as well as for a given full-support prior. We provide counterexamples to underscore the necessity of the assumptions underlying some of our findings, and offer a generalization of the Hirschleifer-Schlee theorem as an application.

Convex Roman Dominating Function in Graphs

Let $G$ be a connected graph. A function $f:V(G)\rightarrow \{0,1,2\}$ is a \textit{convex Roman dominating function} (or CvRDF) if every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$ and $V_1 \cup V_2$ is convex. The weight of a convex Roman dominating function $f$, denoted by $\omega_{G}^{CvR}(f)$, is given by $\omega_{G}^{CvR}(f)=\sum_{v \in V(G)}f(v)$. The minimum weight of a CvRDF on $G$, denoted by $\gamma_{CvR}(G)$, is called the \textit{convex Roman domination number} of $G$. In this paper, we determine the convex Roman domination numbers of some graphs and give some realization results involving convex Roman domination, connected Roman domination, and convex domination numbers.

THE FORCING CONVEX DOMINATION NUMBER OF A GRAPH

Let G be a connected graph and D a minimum convex domination set of G. A subset T ⊆ D is called a forcing subset of D, if D is the unique minimum convex dominating set containing T. A forcing subset for D of minimum cardinality is a minimum forcing subset of D. The forcing convex domination number of D, denoted by γcon(D), is the cardinality of a minimum forcing subset of D. The forcing convex domination number of G, denoted by fγcon(G) and is defined by fγcon(G) = min {fγcon(D)}, where the minimum is taken over all minimum convex dominating sets D in G. Some general properties satisfied by this concepts are studied. The forcing fair dominating number of certain standard graphs are determined. It is shown that for every pair a, b of integers with 0 ≤ a <b, there exists a connected graph G such that fγcon(G) = a and γcon(G) = b.

The convex and weak convex domination number of convex polytopes

The convex and weak convex domination number of convex polytopes

Sharp Khinchin-type inequalities for symmetric discrete uniform random variables

We establish several optimal moment comparison inequalities (Khinchin-type inequalities) for weighted sums of independent identically distributed symmetric discrete random variables which are uniform on sets of consecutive integers. Specifically, we obtain sharp constants for the second moment and any moment of order at least 3 (using convex dominance by Gaussian random variables). In the case of only 3 atoms, we also establish a Schur-convexity result. For moments of order less than 2, we get sharp constants in two cases by exploiting Haagerup’s arguments for random signs.

Weak Convex Domination in Hypercubes

The n-cube Qn is the graph whose vertex set is the set of all n-dimensional Boolean vectors, two vertices being joined if and only if they differ in exactly one coordinate. The n-star graph Sn is a simple graph whose vertex set is the set of all n! permutations of {1, 2, · · · , n} and two vertices α and β are adjacent if and only if α(1)≠β(1) and α(i) ≠β(i) for exactly one i, i≠ 1.
 In this paper we determine weak convex domination number for hypercubes. Also convex, weak convex, m - convex and l1-convex numbers of star and hypercube graphs are determined.

加边对图的弱凸控制数和凸控制数的影响

加边对图的弱凸控制数和凸控制数的影响

Weakly convex and convex domination numbers for generalized Petersen and flower snark graphs

We consider the weakly convex and convex domination numbers for two classes of graphs: generalized Petersen graphs and flower snark graphs. For a given generalized Petersen graph GP(n, k), we prove that if k = 1 and n ≥ 4 then both the weakly convex domination number γwcon(GP(n, k)) and the convex domination number γcon(GP(n, k)) are equal to n. For k ≥ 2 and n ≥ 13, γwcon(GP(n, k)) = γcon(GP(n, k)) = 2n, which is the order of GP(n, k). Special cases for smaller graphs are solved by the exact method. For a flower snark graph Jn, where n is odd and n ≥ 5, we prove that γwcon(Jn) =2n and γcon(Jn) = 4n.

Convex Stochastic Dominance in Bayesian Localization, Filtering, and Controlled Sensing POMDPs

This paper provides conditions on the observation probability distribution in Bayesian localization and optimal filtering so that the conditional mean estimate satisfies convex stochastic dominance. Convex dominance allows us to compare the unconditional mean square error between two optimal Bayesian state estimators over arbitrary time horizons instead of using brute force Monte-Carlo computations. The proof uses two key ideas from microeconomics, namely, integral precision dominance and aggregation of single crossing. The convex dominance result is then used to give sufficient conditions so that the optimal policy of a controlled sensing two-state partially observed Markov decision process (POMDP) is lower bounded by a myopic policy. Numerical examples are presented where the Shannon capacity of the observation distribution using one sensor dominates that of another, and convex dominance holds but Blackwell dominance does not hold. These illustrate the usefulness of the main result in localization, filtering and controlled sensing applications.

Convex dominating sets in maximal outerplanar graphs

In this paper, we study the concept of convex domination in maximal outerplanar graphs. For this class of graphs, we discuss several properties of this domination parameter, in particular, we provide upper bounds on the convex domination number and study effects on the convex domination number when a maximal outerplanar graph is modified by flipping a diagonal. We also propose a linear time algorithm for computing a minimum convex dominating set in a given maximal outerplanar graph. In addition, we consider the concept of convex guard sets in maximal outerplanar graphs and simple polygons.

Convex and weakly convex domination in prism graphs

For a given graph $G=(V,E)$ and permutation $\pi:V\mapsto V$ the prism $\pi G$ of $G$ is defined as follows: $V(\pi G)=V(G)\cup V(G')$, where $G'$ is a copy of $G$, and $E(\pi G)=E(G)\cup E(G')\cup M_{\pi}$, where $M_{\pi}=\{uv': u\in V(G), v=\pi(u)\}$ and $v'$ denotes the copy of $v$ in $G'$. We study and compare the properties of convex and weakly convex dominating sets in prism graphs. In particular, we characterize prism $\gamma_{con}$-fixers and -doublers. We also show that the differences $\gamma_{wcon}(G)-\gamma_{wcon}(\pi G)$ and $\gamma_{wcon}(\pi G) - 2\gamma_{wcon}(G)$ can be arbitrarily large, and that the convex domination number of $\pi G$ cannot be bounded in terms of $\gamma_{con}(G).$

Mean Residual Life Processes and Associated Submartingales

We use Madan-Yor's argument to construct associated submartingales to a class of two-parameter processes that are ordered by the increasing convex dominance. This class includes processes which have MTP$_2$ integrated survival functions. We prove that the integrated survival function of an integrable two-parameter process is MTP$_2$ if and only if it is TP$_2$ in each pair of arguments when the remaining argument is fixed. This result can not be deduced from known results since there are several two-parameter processes whose integrated survival functions do not have interval support. The MTP$_2$ property of certain MRL processes is useful to exhibit numerous other processes having the same property.

Convex and isometric domination of (weak) dominating pair graphs

A set D of vertices in a graph G is a dominating set if every vertex of G, which is not in D, has a neighbor in D. A set of vertices D in G is convex (respectively, isometric), if all vertices in all shortest paths (respectively, all vertices in one of the shortest paths) between any two vertices in D lie in D. The problem of finding a minimum convex dominating (respectively, isometric dominating) set is considered in this paper from algorithmic point of view. For the class of weak dominating pair graphs (i.e., the graphs that contain a dominating pair, which is a pair of vertices x,y∈V(G) such that vertices of any path between x and y form a dominating set), we present an efficient algorithm that finds a minimum isometric dominating set of such a graph. On the other hand, we prove that even if one restricts to weak dominating pair graphs that are also chordal graphs, the problem of deciding whether there exists a convex dominating set bounded by a given arbitrary positive integer is NP-complete. By further restricting the class of graphs to chordal dominating pair graphs (i.e., the chordal graphs in which every connected induced subgraph has a dominating pair) we are able to find a polynomial time algorithm that determines the minimum size of a convex dominating set of such a graph.

$1$-MOVABLE CLIQUE DOMINATING SETS OF A GRAPH

A clique (convex) dominating set S of G is a 1-movable clique dominating set (resp. 1-movable convex dominating set) of G if for every v ∈ S, either S {v} is a clique (resp. convex) dominating set or there exists a vertex u ∈ (V (G) S) ∩ NG(v) such that (S {v}) ∪ {u} is a clique (resp. convex) dominating set of G. The minimum cardinality of a 1-movable clique (resp. 1-movable convex) dominating set of G, denoted by 1 mcl(G) (resp. 1 mcon(G)), is called the 1-movable clique domination number (resp. 1-movable convex domination number) of G. A 1-movable clique dominating set in G with cardinality 1 mcl(G) is called a 1-set of G. This paper aims to characterize the 1-movable clique dominating sets of some graphs including those resulting from the join and composition of two graphs. The corresponding 1-movable clique domination number of the resulting graph is then determined. Further, it is shown that the concepts of 1-movable clique domination and 1-movable convex domination are equivalent.

Weakly convex domination subdivision number of a graph

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.

Convex dominating-geodetic partitions in graphs

The distance d(u,v) between two vertices u and v in a connected graph G is the length of a shortest u-v path in G. A u-v path of length d(u,v) is called u-v geodesic. A set X is convex in G if vertices from all a -b geodesics belong to X for every two vertices a,b?X. A set of vertices D is dominating in G if every vertex of V-D has at least one neighbor in D. The convex domination number con(G) of a graph G equals the minimum cardinality of a convex dominating set in G. A set of vertices S of a graph G is a geodetic set of G if every vertex v ? S lies on a x-y geodesic between two vertices x,y of S. The minimum cardinality of a geodetic set of G is the geodetic number of G and it is denoted by g(G). Let D,S be a convex dominating set and a geodetic set in G, respectively. The two sets D and S form a convex dominating-geodetic partition of G if |D| + |S| = |V(G)|. Moreover, a convex dominating-geodetic partition of G is called optimal if D is a ?con(G)-set and S is a g(G)-set. In the present article we study the (optimal) convex dominating-geodetic partitions of graphs.

Weakly convexity and weakly convex domination in graphs

In this paper, we revisit the weakly convexity in graphs and the weakly convex domination in graphs. Some graphs were constructed with a pre-assigned order, diameter, weakly convexity number, convex number, weakly convex domination number and convex domination number. Mathematics Subject Classication: 05C12

Secure weakly convex domination in graphs

In this paper, we investigate the concept of secure weakly convex domination set of some graphs. We characterized those graphs for which the secure weakly convex domination numbers are 1 and 2. Relations of this parameter with some domination parameters are also observed and a graph is constructed with a preassigned order, weakly convex domination number, secure weakly convex domination number, and secure convex domination number.

Secure convex domination in a graph

Secure convex domination in a graph