分裂图的区间图完全化问题（英文）

The interval graph completion problem on a graph G is to find an added edge set F such that G + F is an interval supergraph with the smallest possible number of edges. The problem has important applications to numerical algebra, V LSI-layout and algorithm graph theory etc; And it has been known to be N P-complete on general graphs. Some classes of special graphs have been investigated in the literatures. In this paper the interval graph completion problem on split graphs is investigated.     

Distance Integral Complete Multipartite Graphs with s=5, 6

Let D(G) = (dij )n×n denote the distance matrix of a connected graph G with order n, where dij is equal to the distance between vertices vi and vj in G. A graph is called distance integral if all eigenvalues of its distance matrix are integers. In 2014, Yang and Wang gave a su?cient and necessary condition for complete r-partite graphs Kp1,p2,··· ,pr =Ka1·p1,a2·p2,··· ,as···ps to be distance integral and obtained such distance integral graphs with s = 1, 2, 3, 4. However distance integral complete multipartite graphs Ka1·p1,a2·p2,··· ,as·ps with s>4 have not been found. In this paper, we find and construct some infinite classes of these distance integral graphs Ka1·p1,a2·p2,··· ,as·ps with s = 5, 6. The problem of the existence of such distance integral graphs Ka1·p1,a2·p2,··· ,as·ps with arbitrarily large number s remains open.     

D(β)-vertex-distinguishing total coloring of graphs

A new concept of the D(β)-vertex-distinguishing total coloring of graphs, i.e., the proper total coloring such that any two vertices whose distance is not larger thanβhave different color sets, where the color set of a vertex is the set composed of all colors of the vertex and the edges incident to it, is proposed in this paper. The D(2)-vertex-distinguishing total colorings of some special graphs are discussed, meanwhile, a conjecture and an open problem are presented.     

THE HILBERT BOUNDARY PROBLEM OF DOUBLY PERIODIC ANALYTIC FUNCTIONS

The doubly periodic Hilbert boundary value problem is discussed in this paper.First,certain kind of integral representations of doubly quasi-periodic analytic functions inmultiplication is established so that the Dirichlet problem of such functions is solved.Then,by the method of regularization,the Hilbert boundary value problem is transferredto such a problem,and it is reduced at length to some Fredholm integral equation.Thenumber of solutions and conditions of solvability as well as the form of the generalsolution are obtained.     

具有s=5,6的距离整完全多部图（英文）

Let D(G) =(d_(ij))_(n×n) denote the distance matrix of a connected graph G with order n, where d_(ij) is equal to the distance between vertices viand vjin G. A graph is called distance integral if all eigenvalues of its distance matrix are integers. In 2014, Yang and Wang gave a sufficient and necessary condition for complete r-partite graphs K_(p1,p2,···,pr)=K_(a1·p1,a2·p2,···,as···ps) to be distance integral and obtained such distance integral graphs with s = 1, 2, 3, 4. However distance integral complete multipartite graphs K_(a1·p1,a2·p2,···,as·ps) with s 4 have not been found. In this paper, we find and construct some infinite classes of these distance integral graphs K_(a1·p1,a2·p2,···,as·ps) with s = 5, 6. The problem of the existence of such distance integral graphs K_(a1·p1,a2·p2,···,as·ps) with arbitrarily large number s remains open.     

ON SPACES OF FRACTAL FUNCTIONS

In this paper we Ointroduce linear-spaces consisting of continuous functions whose graphs are the attactars of a special class of iterated function systems. We show that such spaces are finite dimensional and give the bases of these spaces in an implicit way. Given such a space, we discuss how to obtain a set of knots for whah the Lagrange interpolation problem by the space is uniquely solvable.     

The genus polynomials of cross-ladder digraphs in orientable surfaces

Some results about the genus distributions of graphs are known,but little is known about those of digraphs.In this paper,the method of joint trees initiated by Liu is generalized to compute the embedding genus distributions of digraphs in orientable surfaces.The genus polynomials for a new kind of 4-regular digraphs called the cross-ladders in orientable surfaces are obtained.These results are close to solving the third problem given by Bonnington et al.     

ON CLASSES OF REGULAR GRAPHS WITH CONSTANT METRIC DIMENSION

In this paper,we are dealing with the study of the metric dimension of some classes of regular graphs by considering a class of bridgeless cubic graphs called the flower snarks Jn,a class of cubic convex polytopes considering the open problem raised in [M.Imran et al.,families of plane graphs with constant metric dimension,Utilitas Math.,in press] and finally Harary graphs H 5,n by partially answering to an open problem proposed in [I.Javaid et al.,Families of regular graphs with constant metric dimension,Utilitas Math.,2012,88:43-57].We prove that these classes of regular graphs have constant metric dimension.     

Enumerating reflexible 2-cell embeddings of connected graphs

Two 2-cell embeddings:X → S and j:X → S of a connected graph X into a closed orientable surface S are congruent if there are an orientation-preserving surface homeomorphism h on S and a graph automorphism γ of X such that h = γj.A 2-cell embedding:X → S of a graph X into a closed orientable surface S is described combinatorially by a pair(X;ρ) called a map,where ρ is a product of disjoint cycle permutations each of which is the permutation of the darts of X initiated at the same vertex following the orientation of S.The mirror image of a map(X;ρ) is the map(X;ρ 1),and one of the corresponding embeddings is called the mirror image of the other.A 2-cell embedding of X is reflexible if it is congruent to its mirror image.Mull et al.[Proc Amer Math Soc,1988,103:321-330] developed an approach for enumerating the congruence classes of 2-cell embeddings of graphs into closed orientable surfaces.In this paper we introduce a method for enumerating the congruence classes of reflexible 2-cell embeddings of graphs into closed orientable surfaces,and apply it to the complete graphs,the bouquets of circles,the dipoles and the wheel graphs to count their congruence classes of reflexible or nonreflexible(called chiral) embeddings.     

Minimum congestion spanning trees in bipartite and random graphs

The first problem considered in this article reads: is it possible to find upper estimates for the spanning tree congestion in bipartite graphs, which are better than those for general graphs? It is proved that there exists a bipartite version of the known graph with spanning tree congestion of order n 3 2 , where n is the number of vertices. The second problem is to estimate spanning tree congestion of random graphs. It is proved that the standard model of random graphs cannot be used to find graphs whose spanning tree congestion has order greater than n 3 2 .     

Independent packings in structured graphs

Packing a maximum number of disjoint triangles into a given graph G is NP-hard, even for most classes of structured graphs. In contrast, we show that packing a maximum number of independent (that is, disjoint and nonadjacent) triangles is polynomial-time solvable for many classes of structured graphs, including weakly chordal graphs, asteroidal triple-free graphs, polygon-circle graphs, and interval-filament graphs. These classes contain other well-known classes such as chordal graphs, cocomparability graphs, circle graphs, circular-arc graphs, and outerplanar graphs. Our results apply more generally to independent packings by members of any family of connected graphs. Research of both authors is supported by the Natural Sciences and Engineering Research Council of Canada.     

Some constructions of integral graphs

A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. Integral graphs are very rare and difficult to find. In this article, we introduce some general methods for constructing such graphs. As a consequence, some infinite families of integral graphs are obtained.     

弦图的补图的最小填充

从图论观点讲,最小填充问题就是在一个图G中添加边集F,使得图G的母图G F是一个弦图而且所添边的边数| F|是最小的,其中最小值| F|称为图G的填充数,表示为f( G) .对一般图来说,最小填充问题是NP-困难的,但是对一些特殊图类来说,这个问题是在多项式时间内可解的.本文给出了弦图的补图-G的填充数f(-G) .     

序列平行图的最小填充

起源于稀疏矩阵计算和其它应用领域的一个图G的最小填充问题就是在G中寻找一个边数| F |最小的添加边集F,使得G+F是弦图.这里最小值| F |称为图G的填充数,表示为f(G).对一般图来说,这个问题是NP-困难问题.一些特殊图类的最小填充问题已被研究.本文给出了序列平行图G的最小填充数的具体值.     

On finding augmenting graphs

Method of augmenting graphs is a general approach to solve the maximum independent set problem. As the problem is generally NP-hard, no polynomial time algorithms are available to implement the method. However, when restricted to particular classes of graphs, the approach may lead to efficient solutions. A famous example of this type is the maximum matching algorithm: it finds a maximum matching in a graph G, which is equivalent to finding a maximum independent set in the line graph of G. In the particular case of line graphs, the method reduces to finding augmenting (alternating) chains. Recent investigations of more general classes of graphs revealed many more types of augmenting graphs. In the present paper we study the problem of finding augmenting graphs different from chains. To simplify this problem, we introduce the notion of a redundant set. This allows us to reduce the problem to finding some basic augmenting graphs. As a result, we obtain a polynomial time solution to the maximum independent set problem in a class of graphs which extends several previously studied classes including the line graphs.     

A forbidden subgraph characterization of line-polar bipartite graphs

A graph is polar if the vertex set can be partitioned into A and B in such a way that the subgraph induced by A is a complete multipartite graph and the subgraph induced by B is a disjoint union of cliques. Polar graphs are a common generalization of bipartite, cobipartite, and split graphs. However, recognizing polar graphs is an NP-complete problem in general. This led to the study of the polarity of special classes of graphs such as cographs and chordal graphs, cf. Ekim et al. (2008) [7] and [5]. In this paper, we study the polarity of line graphs and call a graph line-polar if its line graph is polar. We characterize line-polar bipartite graphs in terms of forbidden subgraphs. This answers a question raised in the fist reference mentioned above. Our characterization has already been used to develop a linear time algorithm for recognizing line-polar bipartite graphs, cf. Ekim (submitted for publication) [6].     

The parameterized complexity of the induced matching problem

Given a graph G and an integer k≥0, the NP-complete Induced Matching problem asks whether there exists an edge subset M of size at least k such that M is a matching and no two edges of M are joined by an edge of G. The complexity of this problem on general graphs, as well as on many restricted graph classes has been studied intensively. However, other than the fact that the problem is W[1]-hard on general graphs, little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we provide first-time fixed-parameter tractability results for planar graphs, bounded-degree graphs, graphs with girth at least six, bipartite graphs, line graphs, and graphs of bounded treewidth. In particular, we give a linear-size problem kernel for planar graphs.     

Cover-incomparability graphs and chordal graphs

The problem of recognizing cover-incomparability graphs (i.e. the graphs obtained from posets as the edge-union of their covering and incomparability graph) was shown to be NP-complete in general [J. Maxová, P. Pavlíkova, A. Turzík, On the complexity of cover-incomparability graphs of posets, Order 26 (2009) 229-236], while it is for instance clearly polynomial within trees. In this paper we concentrate on (classes of) chordal graphs, and show that any cover-incomparability graph that is a chordal graph is an interval graph. We characterize the posets whose cover-incomparability graph is a block graph, and a split graph, respectively, and also characterize the cover-incomparability graphs among block and split graphs, respectively. The latter characterizations yield linear time algorithms for the recognition of block and split graphs, respectively, that are cover-incomparability graphs.     

Obstructions for acyclic local tournament orientation completions

The orientation completion problem for a fixed class of oriented graphs asks whether a given partially oriented graph can be completed to an oriented graph in the class. Orientation completion problems have been studied recently for several classes of oriented graphs, including local tournaments. Local tournaments are intimately related to proper circular-arc graphs and proper interval graphs. In particular, proper interval graphs are precisely those which can be oriented as acyclic local tournaments. In this paper we determine all obstructions for acyclic local tournament orientation completions. These are in a sense minimal partially oriented graphs that cannot be completed to acyclic local tournaments. Our results imply that a polynomial time certifying algorithm exists for the acyclic local tournament orientation completion problem.