Strong clique trees,neighborhood trees,and strongly chordal graphs

Maximal complete subgraphs and clique trees are basic to both the theory and applications of chordal graphs. A simple notion of strong clique tree extends this structure to strongly chordal graphs. Replacing maximal complete subgraphs with open or closed vertex neighborhoods discloses new relationships between chordal and strongly chordal graphs and the previously studied families of chordal bipartite graphs, clique graphs of chordal graphs (dually chordal graphs), and incidence graphs of biacyclic hypergraphs. © 2000 John Wiley & Sons, Inc. J. Graph Theory 33: 151–160, 2000     

Domination subdivision numbers of trees

A set S of vertices of a graph G=(V,E) is a dominating set if every vertex of V(G)?S is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G. The domination subdivision number  is the minimum number of edges that must be subdivided in order to increase the domination number. Velammal showed that for any tree T of order at least 3, . In this paper, we give two characterizations of trees whose domination subdivision number is 3 and a linear algorithm for recognizing them.     

On domination and reinforcement numbers in trees

The reinforcement number of a graph is the smallest number of edges that have to be added to a graph to reduce the domination number. We introduce the k-reinforcement number of a graph as the smallest number of edges that have to be added to a graph to reduce the domination number by k. We present an O(k²n) dynamic programming algorithm for computing the maximum number of vertices that can be dominated using γ(G)-k dominators for trees. A corollary of this is a linear-time algorithm for computing the k-reinforcement number of a tree. We also discuss extensions and related problems.     

满足开集条件的自相似集的Lipschitz等价

固定r∈(0,1)及整数N≥2,设E和E'为由N个形如S(x)=±rx+b的压缩映射所生成的自相似集.设开集条件对于E及E'成立,并且所对应的开集为开区间,证明了E和E'Lipschitz等价.     

An optimal locating-dominating set in the infinite triangular grid

Assume that G=(V,E) is an undirected graph, and C⊆V. For every v∈V, we denote by I(v) the set of all elements of C that are within distance one from v. If all the sets I(v) for v∈V?C are non-empty, and pairwise different, then C is called a locating-dominating set. The smallest possible density of a locating-dominating set in the infinite triangular grid is shown to be .     

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.     

A note on balance vertices in trees

The notion of balanced bipartitions of the vertices in a tree T was introduced and studied by Reid (Networks 34 (1999) 264). Reid proved that the set of balance vertices of a tree T consists of a single vertex or two adjacent vertices. In this note, we give a simple proof of that result.     

Median orders of tournaments: A tool for the second neighborhood problem and Sumner's conjecture

We give a short constructive proof of a theorem of Fisher: every tournament contains a vertex whose second outneighborhood is as large as its first outneighborhood. Moreover, we exhibit two such vertices provided that the tournament has no dominated vertex. The proof makes use of median orders. A second application of median orders is that every tournament of order 2n − 2 contains every arborescence of order n > 1. This is a particular case of Sumner's conjecture: every tournament of order 2n − 2 contains every oriented tree of order n > 1. Using our method, we prove that every tournament of order (7n − 5)/2 contains every oriented tree of order n. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 244–256, 2000     

Characterizations of trees with equal paired and double domination numbers

A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching, and a double dominating set is a dominating set that dominates every vertex of G at least twice. We show that for trees, the paired-domination number is less than or equal to the double domination number, solving a conjecture of Chellali and Haynes. Then we characterize the trees having equal paired and double domination numbers.     

Extremal problems for chromatic neighborhood sets

The chromatic neighborhood sequence of a graph G is the list of the chromatic numbers of the subgraphs induced by the neighborhoods of the vertices. We study the maximum multiplicity of this sequence, proving, amongst other things, that if a chromatic neighborhood sequence has t distinct values, the largest value being d_t, then there is a value with multiplicity at least . This bound is asymptotically tight. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 68–74, 2002     

Remarks on the second neighborhood problem

The second neighborhood conjecture of Seymour asserts that for any orientation G = (V,E), there exists a vertex υ ∈ V so that |N⁺(υ)| ≤ |N⁺⁺(υ)|. The conjecture was resolved by Fisher for tournaments. In this article, we prove the second neighborhood conjecture for several additional classes of dense orientations. We also prove some approximation results, and reduce an asymptotic version of the conjecture to a finite case. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 208–220, 2007     

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.     

Hamilton连通性和邻域并条件

设G＝（V，E）为简单图，δ为图G的最小度，1987年Faudree等人给出NC＝min{|N(x)∪N(y)‖x,y∈V(G),xy∈N(G)}，有关文献曾研究3连通的H连通图，本文进一步得到：若G是n阶2连通图，且NC≥n-δ，则G除几个图外均是H连通图，从而，完成了邻并条件的H连通图问题。     

A variable neighborhood search based algorithm for finite-horizon Markov Decision Processes

This paper considers the application of a variable neighborhood search (VNS) algorithm for finite-horizon (H stages) Markov Decision Processes (MDPs), for the purpose of alleviating the “curse of dimensionality” phenomenon in searching for the global optimum. The main idea behind the VNSMDP algorithm is that, based on the result of the stage just considered, the search for the optimal solution (action) of state x in stage t is conducted systematically in variable neighborhood sets of the current action. Thus, the VNSMDP algorithm is capable of searching for the optimum within some subsets of the action space, rather than over the whole action set. Analysis on complexity and convergence attributes of the VNSMDP algorithm are conducted in the paper. It is shown by theoretical and computational analysis that, the VNSMDP algorithm succeeds in searching for the global optimum in an efficient way.     

Hamiltonian graphs involving neighborhood unions

Dirac proved that a graph G is hamiltonian if the minimum degree , where n is the order of G. Let G be a graph and . The neighborhood of A is for some . For any positive integer k, we show that every (2k ? 1)‐connected graph of order n ≥ 16k³ is hamiltonian if |N(A)| ≥ n/2 for every independent vertex set A of k vertices. The result contains a few known results as special cases. The case of k = 1 is the classic result of Dirac when n is large and the case of k = 2 is a result of Broersma, Van den Heuvel, and Veldman when n is large. For general k, this result improves a result of Chen and Liu. The lower bound 2k ? 1 on connectivity is best possible in general while the lower bound 16k³ for n is conjectured to be unnecessary. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 83–100, 2006     

Refinable maps and generalized absolute neighborhood retracts

This paper studies some properties which are preserved by refinable maps when the domain is a certain kind of generalized ANR called a quasi-ANR. Among those properties are the properties of being a quasi-ANR and the disjoint n-cell property.An alternative characterization of quasi-ANR's as surjective approximative neighborhood extensors is also obtained in Theorems 3 and 4. These characterizations are analogs of similar characterizations in ANR theory.