IS γα≤ 5 FOR GRAPHS WHICH HAVE DIAMETER TWO？

A subset of S of the vertex set of a graph G is called acyclic if the subgraph it induces in G contains no cycles. S is called an acyclic dominating set of G if it is both acyclic and dominating. The minimum cardinality of an acyclic dominating set, denoted by γα(G), is called the acyclic domination number of G. S. M. Hedetniemi et al. on 2000 introduced the concept of acyclic domination and posed the following open problem: Is γα(G) ≤ δ(G) for any graph whose diameter is two? In this paper, we give a counterexample which disproves the problem.     

CYCLES CONTAINING MANY VERTICES OF SUBSETS IN GRAPHS WITH LARGE DEGREE SUMS AND NEIGHBORHOOD UNIONS

1. ResultsWe use[IJ for terminology and notations not defined here and consider simple graph only.Let G be a graph of order n and X C V(G). A graph G is called 1--tough if w(G\S) 5 ISIfor any S g V(G) with w(G\S) > 1, where w(G\S) denotes the number of components ofgraph G\S. A cycle C of G is called X-longest if no cycle of G contains more venices of Xthan C, and by c(X) we denote the number of venices of X in an X-longest cycle. A cycle Cof G is called X-dominating if all neigh…     

A NOTE ON CONNECTED FACTORS IN CLAW-FREE GRAPHS

1. IntroductionAll graphs considered here are finite and undirected, without loops or multiple edges. Thevertex set, edge set and the degree of a vertex v of a graph G are denoted, respectively, byV(G),E(G) and dG(v). For a set S G V(G) the subgraph of G induced by S is denoted byG[S].Let G be a graph, j and g positive integer Valued functions on V(G) with f(v) S g(v) 5dG(u) for all v 6 V(G). A spanmng SUbgraph F Of G is called an [f,g]-factor Of G if forall v E V(G), f(v) 5 dF(v…     

A FURTHER GENERALIZATION OF JUNG''''S THEOREM

Let G be a graph of order n. We define the distance between two vertices u andv in G, denoted by d(u, v), as the minimum value of the lengths of all u-v paths. We writeσ_k(G)=min{∑_i=1~k d(v_i)|{v_1, v_2,…, v_k} is an independent set in G} and NC2(G)=min {|N(u)∪N(v)| | d(u, v)=2}. We denote by ω(G) the number of components of agraph G. A graph G is called 1-tough if ω(G\S)≤|S| for every subset S of V(G) withω(G\S)>l. By c(G) we denote the length of the longest cycle in G; in particular, G iscalled a Hamiltonian graph if c(G)=n. H.A. Jung proved that every 1-tough graphwith order n≥11 and σ2≥n-4 is Hamiltonian. We generalize it further as follows: ifG is a 1-tough graph and σ3(G)≥n, then c(G)≥min {n,2NC2(G)+4}. Thus, theconjecture of D. Bauer, G. Fan and H.J. Veldman in [2] is completely solved.     

(4m, m)-CHOOSABILITY OF PLANE GRAPHS

1 IntroductionAll graphs considered in tabs paper are abate and sable. Undecked notations and termalnologies can be found in [if.Let G = (V, E,F) be a plase graph, where V, E and F denote the s6t of venices, edgesand faces of G, respectively. We use NG(v) and dG(v) to denote the set and nUmber of venicesadjacent to a vertex yi respectively, and use 6(G) to denote the inininuun degree of G. A vertexv is called a k-vertex if v has degree k. A face of a plane graph is said to be incident w…     

ON THE INVERSE MINIMUM SPANNING TREE PROBLEM WITH MINIMUM NUMBER OF PERTURBED EDGES

Let G=be a network with the vertex set V,the edge set E and the length vector L, andlet T~* be a prior determined spanning tree of G. The inverse minimum spanning tree problem withminimum number of perturbed edges is to perturb the length vector L to L+δ, such that T~* is one ofminimum spanning trees under the length vector L+δ and the number of perturbed edges is minimum.This paper establishes a mathematical model for this problem and transforms it into a minimumvertex covering problem in a bipartite graph G_0, a path-graph. Thus a strongly polynomial algorithmwith time complexity O(mn~2) can be designed by using Hungarian method.     

EQUITABLE COLORINGS OF LINE GRAPHS AND COMPLETE r-PARTITE GRAPHS

1. IntroductionLet G be a graph with vertex set V(G), edge set E(G), maximum degree A(G), minimumdegree 8(G), vertex chromatic number X(G), and edge chromatic number X'(G). G is equitablyk-colorable if V(G) can be pajrtitioned icao k independent sets VI, V2,'', Vk such that llKI III 1 5 1 for all i and j. The such smallest integer k as above is called the equitable chromaticnumber of G, denoted by X= (G). Similaxly we can define the equitable edge chromatic numberof a graph G and d…     

THE NORMALITY OF CAYLEY GRAPHS OF FINITE ABELIAN GROUPS WITH VALENCY 5

1. IntroductionLet G be a finite group and S a subset of G not colltaining the idelltity elemellt 1. TheCayley digraph X ~ Cay(G, S) of G on S is defined byV(X) = G, E(X) ~ {(g, sg)jg E G, s E S}.Immediately from the definition there are three obvious facts: (1) Ant(X), the automorphismgroup of X, contains the right regular representation GR of G; (2) X is connected if and onlyif G = (S}, (3) X is an undirected graph (by coalescing each pair (g, sg) and (sg, g) of directededges int…     

PACKING A TREE OF ORDER p WITH A （p,p＋1）—GRAPH

Let G1 and G2 be two graphs of the same order,If G1 is isomorphic to a spanning subgraph of the complement of G2,then we say that G1 and G2 are packable.A graph G is called a (p,m)-graph if G has p vertices and m edges.The main purpose of this paper is to present a necessary and sufficient condition for a tree of order p and a (p,p 1)-graph to be packable.     

ON ACYCLIC AND CYCLIC HYPERGRAPHS

So far, the acyclic hypergraph has two different definitions. One is based on the cyclomatic number of the hypergraph, whereas the other arises from the acyclic schema of the relational database in the computer science. In this paper, it is first proved that these two definitions coincide with each other completely. Then we prove that a hypergraph H is not acyclic, or cyclic, if and only if it contains a special partial hypergraph named hypercircuit. In addition, we show that H has l(H) different hypercircuits, where l(H) is a parameter used to decide whether H is acyclic or cyclic.     

ON THE LEAST EIGENVALUE OF A GRAPH

Let G be a simple graph with n vertices and λ_n(G) be the least eigenvalue of G.In this paper, we show that, if G is connected but not complete, then λ_n(G)≤λ_n(K_(n-1)~1)and the equality holds if and only if G K_(n-1)~1, where K_(n-1)~1, is the graph obtained by thecoalescence of a complete graph K_(n-1) of n-1 vertices with a path P_2 of length one of itsvertices.     

FORBIDDEN SUBGRAPHS,DEGREES,NEIGHBORHOOD UNIONS AND CIRCUMFERENCES OF GRAPHS

Combining forbidden subgraphs with degree restrictions and neighborhood unionrestrictions,respectively,we prove the following results:(1) Let G be a 2-connected graph of order n,and 3≤c≤n.If for each induced subgraphL of order four of G(?)|V_1(L)∩S_c|≥2 if L≌K_(1,3),and |V(L)∩S_c|≥1 if L≌P_4,then thecircumference of G is at least c,where V_1(L)is the set of vertices with degree 1 of L,S_c isthe set of vertices with degree at least c/2 of G and P_4 is a path of order 4.(2) Let G be a 2-connected graph of order n,and n≥s+2.If for each induced subgraphL of G isomorphic to K_(1,3)or P_4,d_L(u,v)=2(?)|N(u)∪N(v)|≥s,then the circumferencec (G) of G is at least s+2.Moreover,if n≥s+3 and s is odd,then c(G)≥s+3.     

IMPLICIT DEGREES AND CHVATAL'S CONDITION FOR HAMILTONICITY

Based on the ideas in[9],an integer d~0(v),called the implicit degree of v whichsatisfies d~0(v)≥d(v),is associated with each vertex v of a graph G.It is proved that if theimplicit degree sequence d_1~0,d_2~0,…,d_n~0(where d_1~0≤d_2~0≤…≤d_n~0)of a simple graph G on n≥3vertices satisfiesd_i~0≤i相似文献   

CIRCUMFERENCES OF CLAW-FREE GRAPHS

The paper was done when the author studied for Ph. D at Institute of Systems Science, Academia Sinica.Let G be a graph. The circumference c(G) of G is the maximum length of cycles in G. Gis claw-free if it does not contain a copy of K1,3 as an induced subgraph. There have been agreat many results in recent years dealing with claw-free e graphs. In this paper we discuss thecircumferences of claw-free graphs, and obtain a theorem which generalizes a result of Tian,Wu and Liu to k-connecte…     

Hybrid hierarchical trajectory planning for a fixed-wing UCAV performing air-to-surface multi-target attack

This paper considers the problem of generating a flight trajectory for a single fixed-wing unmanned combat aerial vehicle (UCAV) performing an air-to-surface multi-target attack (A/SMTA) mission using satellite-guided bombs. First, this problem is formulated as a variant of the traveling salesman problem (TSP), called the dynamic-constrained TSP with neighborhoods (DCTSPN). Then, a hierarchical hybrid approach, which partitions the planning algorithm into a roadmap planning layer and an optimal control layer, is proposed to solve the DCTSPN. In the roadmap planning layer, a novel algorithm based on an updatable probabilistic roadmap (PRM) is presented, which operates by randomly sampling a finite set of vehicle states from continuous state space in order to reduce the complicated trajectory planning problem to planning on a finite directed graph. In the optimal control layer, a collision-free state-to-state trajectory planner based on the Gauss pseudospectral method is developed, which can generate both dynamically feasible and optimal flight trajectories. The entire process of solving a DCTSPN consists of two phases. First, in the offline preprocessing phase, the algorithm constructs a PRM, and then converts the original problem into a standard asymmetric TSP (ATSP). Second, in the online querying phase, the costs of directed edges in PRM are updated first, and a fast heuristic searching algorithm is then used to solve the ATSP. Numerical experiments indicate that the algorithm proposed in this paper can generate both feasible and near-optimal solutions quickly for online purposes.     

A rough set approach to feature selection based on scatter search metaheuristic

Rough set theory is an effective method to feature selection, which has recently fascinated many researchers. The essence of rough set approach to feature selection is to find a subset of the original features. It is, however, an NP-hard problem finding a minimal subset of the features, and it is necessary to investigate effective and efficient heuristic algorithms. This paper presents a novel rough set approach to feature selection based on scatter search metaheuristic. The proposed method, called scatter search rough set attribute reduction （SSAR）, is illustrated by 13 well known datasets from UCI machine learning repository. The proposed heuristic strategy is compared with typical attribute reduction methods including genetic algorithm, ant colony, simulated annealing, and Tabu search. Computational results demonstrate that our algorithm can provide efficient solution to find a minimal subset of the features and show promising and competitive performance on the considered datasets.     

TO IMPROVE THE COMMUNICATION DELAY BYUPGRADING NODES IN A CONTINUOUS VERSION

In this paper, we consider a network communication delay improvement prob-lem, which is to upgrade nodes in a network with minimum cost such that the communication delay between any two nodes of the network is below a pre-specific level. In the upgrading model, the improvement by upgrading one node is a continuous variable, and the cost incurred by such an upgrading is a linear function of the improvement. We show that achieving an approximation ratio β In(|V|) for the problem is NP-hard for some constant β＞0 even if the underlying network is a bipartite graph. But if the underlying network is restricted as a tree, we show that it can be solved in a strongly polynomial time.     

Several Sufficient Conditions for Class I

1　IntroductionIn1 964 ,the Soviet Union mathematician V.G.Vizing made a breakthrough resulton edgescoloring of graph:if G is a simple graph whose maximum degree isΔ ,then the edges of Gcan be normally colored withΔorΔ 1 colors.From this resultthe problem of classifyingsimple graphs arises,that is,what types of simple graph can be normally colored byΔcolors. Accordingly,there exists a similar problem of complex graphs. Solving theclassifying problem thoroughly means solving the4 -colo…     

THE MATRIX ITERATION ALGORITHM SOLVING AN ENUMERATION PROBLEM ON BACTERIAL COMPLETE GENOMES

Given an alphabet E and a finite minimal set B of forbidden words, a combinatorial enumeration problem on bacterial complete genomes is transformed to enumerating strings of a given length which do not, contain any string in B as their substrings. Prom the fact that a string in the language is equivalent to a path in the corresponding graph, we have obtained a polynomial time algorithm by modifying the power of the adjacency matrix in the graph.     

DEGREE CONDITIONS FOR GRAPHS TO BE FRACTIONAL （a, b, n）-CRITICAL GRAPHS

Let G be a graph, and a and b be integers with a ≤ b. A graph G is called a fraetional （a, b, n）-critical graph if after any n vertices of G are deleted the remaining subgraph has a fractional [a, b]-factor. In this paper two degree conditions for graphs to be fractional （a, b, n）-eritical graphs are presented, and the degree conditions are sharp in some sense.