Broadcasts and domination in trees

A broadcast on a graph G is a function f:V→Z⁺∪{0}. The broadcast number of G is the minimum value of ∑_v∈Vf(v) among all broadcasts f for which each vertex of G is within distance f(v) from some vertex v with f(v)≥1. This number is bounded above by the radius and the domination number of G. We show that to characterize trees with equal broadcast and domination numbers it is sufficient to characterize trees for which all three of these parameters coincide.     

树的3-彩虹控制数的上界

对树的3-彩虹控制数进行研究,首先用构造法找到直径较小的树的3-彩虹控制数的上界.再通过分类讨论思想和数学归纳法得到一般的阶n大于等于5的树的3-彩虹控制数的上界.     

The limit case of a domination property

The domination number γ(G) of a connected graph G of order n is bounded below by(n+2-e(G))/ 3 , where (G) denotes the maximum number of leaves in any spanning tree of G. We show that (n+2-e(G))/ 3 = γ(G) if and only if there exists a tree T ∈ T ( G) ∩ R such that n1(T ) = e(G), where n1(T ) denotes the number of leaves of T1, R denotes the family of all trees in which the distance between any two distinct leaves is congruent to 2 modulo 3, and T (G) denotes the set composed by the spanning trees of G. As a consequence of the study, we show that if (n+2-e(G))/ 3 = γ(G), then there exists a minimum dominating set in G whose induced subgraph is an independent set. Finally, we characterize all unicyclic graphs G for which equality (n+2-e(G))/ 3= γ(G) holds and we show that the length of the unique cycle of any unicyclic graph G with (n+2-e(G))/ 3= γ(G) belongs to {4} ∪ {3 , 6, 9, . . . }.     

Weakly connected domination stable trees

A dominating set D ⊆ V(G) is a weakly connected dominating set in G if the subgraph G[D] _w = (N _G [D], E _w ) weakly induced by D is connected, where E _w is the set of all edges having at least one vertex in D. Weakly connected domination number _γw (_G) of a graph G is the minimum cardinality among all weakly connected dominating sets in G. A graph G is said to be weakly connected domination stable or just _γw -stable if _γw (G) = _{γ w} (G + e) for every edge e belonging to the complement Ḡ of G. We provide a constructive characterization of weakly connected domination stable trees.      

The distance-domination numbers of trees

The P-center problem is to locate P centers in a graph G so that the maximum distance between centers and non-centers is minimized. A related problem is to determine the maximum number of vertices that can be “covered” (within a distance of α) by a vertex set of cardinality P in G. In this paper we describe an O(n³P) algorithm which solves the maximum coverage problem on trees. We also apply the same idea to solve the P-median problem on trees.     

Infinite versus finite graph domination

We raise the following general problem: Which structural properties of dominating subgraphs in finite graphs remain valid for infinite graphs? Positive and negative results are presented.     

Total outer-connected domination numbers of trees

Let G=(V,E) be a graph without an isolated vertex. A set DV(G) is a total dominating set if D is dominating, and the induced subgraph G[D] does not contain an isolated vertex. The total domination number of G is the minimum cardinality of a total dominating set of G. A set DV(G) is a total outer-connected dominating set if D is total dominating, and the induced subgraph G[V(G)−D] is a connected graph. The total outer-connected domination number of G is the minimum cardinality of a total outer-connected dominating set of G. We characterize trees with equal total domination and total outer-connected domination numbers. We give a lower bound for the total outer-connected domination number of trees and we characterize the extremal trees.     

Broadcasting from multiple originators

We begin an investigation of broadcasting from multiple originators, a variant of broadcasting in which any k vertices may be the originators of a message in a network of n vertices. The requirement is that the message be distributed to all n vertices in minimum time. A minimumk-originator broadcast graph is a graph on n vertices with the fewest edges such that any subset of k vertices can broadcast in minimum time. B_k(n) is the number of edges in such a graph. In this paper, we present asymptotic upper and lower bounds on B_k(n). We also present an exact result for the case when . We also give an upper bound on the number of edges in a relaxed version of this problem in which one additional time unit is allowed for the broadcast.     

Upper bounds on the upper signed total domination number of graphs

Let G=(V,E) be a graph. A function f:V→{−1,+1} defined on the vertices of G is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. A signed total dominating function f is minimal if there does not exist a signed total dominating function g, f≠g, for which g(v)≤f(v) for every v∈V. The weight of a signed total dominating function is the sum of its function values over all vertices of G. The upper signed total domination number of G is the maximum weight of a minimal signed total dominating function on G. In this paper we present a sharp upper bound on the upper signed total domination number of an arbitrary graph. This result generalizes previous results for regular graphs and nearly regular graphs.     

Total restrained domination in trees

Let G=(V,E) be a graph. A set S⊆V is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of V-S is adjacent to a vertex in V-S. The total restrained domination number of G, denoted by γ_tr(G), is the smallest cardinality of a total restrained dominating set of G. We show that if T is a tree of order n, then . Moreover, we show that if T is a tree of order , then . We then constructively characterize the extremal trees T of order n achieving these lower bounds.     

Strong equality of domination parameters in trees

We study the concept of strong equality of domination parameters. Let P₁ and P₂ be properties of vertex subsets of a graph, and assume that every subset of V(G) with property P₂ also has property P₁. Let ψ₁(G) and ψ₂(G), respectively, denote the minimum cardinalities of sets with properties P₁ and P₂, respectively. Then ψ₁(G)ψ₂(G). If ψ₁(G)=ψ₂(G) and every ψ₁(G)-set is also a ψ₂(G)-set, then we say ψ₁(G) strongly equals ψ₂(G), written ψ₁(G)≡ψ₂(G). We provide a constructive characterization of the trees T such that γ(T)≡i(T), where γ(T) and i(T) are the domination and independent domination numbers, respectively. A constructive characterization of the trees T for which γ(T)=γ_t(T), where γ_t(T) denotes the total domination number of T, is also presented.     

Optimal broadcast domination in polynomial time

Broadcast domination was introduced by Erwin in 2002, and it is a variant of the standard dominating set problem, such that different vertices can be assigned different domination powers. Broadcast domination assigns an integer power f(v)?0 to each vertex v of a given graph, such that every vertex of the graph is within distance f(v) from some vertex v having f(v)?1. The optimal broadcast domination problem seeks to minimize the sum of the powers assigned to the vertices of the graph. Since the presentation of this problem its computational complexity has been open, and the general belief has been that it might be NP-hard. In this paper, we show that optimal broadcast domination is actually in P, and we give a polynomial time algorithm for solving the problem on arbitrary graphs, using a non-standard approach.     

Paired bondage in trees

Let G=(V,E) be a graph with δ(G)≥1. A set D⊆V is a paired dominating set if D is dominating, and the induced subgraph 〈D〉 contains a perfect matching. The paired domination number of G, denoted by γ_p(G), is the minimum cardinality of a paired dominating set of G. The paired bondage number, denoted by b_p(G), is the minimum cardinality among all sets of edges E^′⊆E such that δ(G−E^′)≥1 and γ_p(G−E^′)>γ_p(G). We say that G is a γ_p-strongly stable graph if, for all E^′⊆E, either γ_p(G−E^′)=γ_p(G) or δ(G−E^′)=0. We discuss the basic properties of paired bondage and give a constructive characterization of γ_p-strongly stable trees.     

Constructive characterizations of (γp,γ)- and (γp,γpr)-trees

Let G = (V,E) be a graph without isolated vertices.A set S V is a domination set of G if every vertex in V - S is adjacent to a vertex in S,that is N[S] = V.The domination number of G,denoted by γ(G),is the minimum cardinality of a domination set of G.A set S C V is a paired-domination set of G if S is a domination set of G and the induced subgraph G[S] has a perfect matching.The paired-domination number,denoted by γpr(G),is defined to be the minimum cardinality of a paired-domination set S in G.A subset S V is a power domination set of G if all vertices of V can be observed recursively by the following rules: (i) all vertices in N[S] are observed initially,and (ii) if an observed vertex u has all neighbors observed except one neighbor v,then v is observed (by u).The power domination number,denoted by γp(G),is the minimum cardinality of a power domination set of G.In this paper,the constructive characterizations for trees with γp = γ and γpr = γp are provided respectively.     

关于树T的γ/γc的界

控制γ和连通控制数γc是图的两个重要的控制参数，本文通过对树中的点进行恰当分类，给出了树中的γ/γc值的最好界，为刻画单圈图和双圈图中γ/γc值的界打下良好的基础。     

Constructive characterizations of （γp, γ）- and （γp, γpr）-trees

Let G =(V,E) be a graph without isolated vertices.A set S  V is a domination set of G if every vertex in V -S is adjacent to a vertex in S,that is N[S] = V .The domination number of G,denoted by γ(G),is the minimum cardinality of a domination set of G.A set S  V is a paired-domination set of G if S is a domination set of G and the induced subgraph G[S]has a perfect matching.The paired-domination number,denoted by γpr(G),is defined to be the minimum cardinality of a paired-domination set S in G.A subset S  V is a power domination set of G if all vertices of V can be observed recursively by the following rules:(i) all vertices in N[S] are observed initially,and(ii) if an observed vertex u has all neighbors observed except one neighbor v,then v is observed(by u).The power domination number,denoted by γp(G),is the minimum cardinality of a power domination set of G.In this paper,the constructive characterizations for trees with γp = γ and γpr = γp are provided respectively.     

Constructive characterizations of （γp, γ）- and （γp, γpr）-trees

Let G = （V, E） be a graph without isolated vertices. A set S lohtain in V is a domination set of G if every vertex in V - S is adjacent to a vertex in S, that is N[S] = V. The domination number of G, denoted by γ（G）, is the minimum cardinality of a domination set of G. A set S lohtain in V is a paired-domination set of G if S is a domination set of G and the induced subgraph G[S] has a perfect matching. The paired-domination number, denoted by γpr（G）, is defined to be the minimum cardinality of a paired-domination set S in G. A subset S lohtain in V is a power domination set of G if all vertices of V can be observed recursively by the following rules： （i） all vertices in N[S] are observed initially, and （ii） if an observed vertex u has all neighbors observed except one neighbor v, then v is observed （by u）. The power domination number, denoted by γp（G）, is the minimum cardinality of a power domination set of G. In this paper, the constructive characterizations for trees with γp=γ and γpr = γp are provided respectively.     

Ordered trees and the inorder traversal

In this paper ordered trees are studied with respect to the inorder traversal. New decompositions of ordered trees are introduced and used to obtain enumeration results according to various parameters. Furthermore, the set of all ordered trees with prescribed degree sequence according to the inorder is studied with the aid of Dyck paths. This set is constructed and its cardinal number is evaluated recursively.     

A minimum broadcast graph on 26 vertices

Broadcasting is the process of information dissemination in a communication network in which a message, originated by one member, is transmitted to all members of the network. A broadcast graph is a graph which permits broadcasting from any originator in minimum time. The broadcast function B(n) is the minimum number of edges in any broadcast graph on n vertices. In this paper, we construct a broadcast graph on 26 vertices with 42 edges to prove B(26) = 42.