The Ferry Cover Problem on Regular Graphs and Small-Degree GraphsThe Ferry Cover Problem on Regular Graphs and Small-Degree Graphs

The ferry problem may be viewed as generalizations of the classical wolf-goatcabbage puzzle. The ferry cover problem is to determine the minimum required boat capacity to safely transport n items represented by a conflict graph. The Alcuin number of a conflict graph is the smallest capacity of a boat for which the graph possesses a feasible ferry schedule. In this paper the authors determine the Alcuin number of regular graphs and graphs with maximum degree at most five.     

平面图的Alcuin数

广义渡河问题是一类重要的组合优化问题,它是经典的狼-羊-卷心菜游戏的推广。冲突图是一个图,这个图的任意两个点所代表的物品不相容时(例如,狼和羊代表的物品不相容),则在这两个点之间连结一条边。渡河覆盖问题的目的是确定冲突图全部点所代表的物品从河的一岸安全地摆渡到河的对岸时所需船的最小容量,而冲突图的Alcuin数定义这个最小容量。本文讨论了平面图的Alcuin数, 给出了该类图Alcuin数的完全刻画。     

覆盖数不超过3的图上渡河问题

1000多年前,英国著名学者Alcuin曾提出一个古老的渡河问题,即狼、羊和卷心菜的渡河问题。2006年,Prisner把该问题推广到任意的冲突图上,考虑了一类情况更一般的渡河运输问题。所谓冲突图是指一个图G=(V,E),这里V代表某些物品的集合,V中的两个点有边连结当且仅当这两个点是冲突的,即在无人监管的情况下不允许留在一起的点。图G=(V,E)的一个可行运输方案是指在保证不发生任何冲突的前提下,把V的点所代表的物品全部摆渡到河对岸的一个运输方案。图G的Alcuin数定义为它存在可行运输方案时所需船的最小容量。本文讨论了覆盖数不超过3的连通图的Alcuin数,给出了该类图Alcuin数的完全刻画。     

最大度为5的图的Alcuin数

1000多年前,英国著名学者Alcuin曾提出过一个古老的渡河问题,即狼、羊和卷心菜的渡河问题.最近,Prisner和Csorba等人把这一问题推广到任意的"冲突图"G=(V,E)上,考虑了一类情况更一般的运输计划问题.现在监管者欲运输V中的所有"物品/点"渡河,这里V的两个点邻接当且仅当这两个点为冲突点.冲突点是指不能在无人监管的情况下留在一起的点.特别地,Alcuin渡河问题可转化成"冲突路"P_3上是否存在可行运输方案问题.图G的Alcuin数是指图G具有可行运输方案(即把V的点代表的"物品"全部运到河对岸)时船的最小容量.最大度为5且覆盖数至少为5的图和最大度Δ(G)≤4且覆盖数不小于Δ(G)-1的图的Alcuin数已经被确定.本文给出最大度为4且覆盖数不超过2和最大度为5且覆盖数不超过4的图的Alcuin数.至此,最大度不超过5的图的Alcuin数被完全确定.     

超图的Alcuin数与其横贯数的关系

1000多年前, 英国著名学者Alcuin曾提出过一个古老的渡河问题, 即狼、羊和卷心菜的渡河问题. 最近, Prisner和Csorba等考虑了一般``冲突图"上的渡河问题. 将这一问题推广到超图$H=(V,\mathcal{E})$\,上, 考虑一类情况更一般的运输计划问题. 现在监管者 欲运输超图中的所有点\,(代表``items")\,渡河, 这里$V$的点子 形成超边 当且仅当这些点代表的``items"在无人监管的情况下不能留在一起. 超图$H$的Alcuin数是指超图$H$具有可行运输方案\,(即把$V$的点代表的``items" 全部运到河对岸)\,时船的最小容量. 给出了 $r$-一致完全二部超图和它的伴随超图, 以及$r$-一致超图的Alcuin数, 同时证明了判断$r$-一致超图是否为小船图是NP 困难的.     

图的最小完整度

主要研究了图的完整度,并给出若干关于完整度的结果. 对于所有的顶点数和边数都给定的连通图类,如何确定该图类中完整度最小的图. 同时研究了对于顶点数和完整度都给定的连通图类,如何确定该图类中边数最多的图的问题. 这些结果为图的最小完整度的优化设计提供了理论和方法.     

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) .     

无向路图和块图上的混合控制

图G=(V,E)的一个混合控制集是一个满足如下条件的集合DV∪E:不在D中的每个点或每条边都相邻或关联于D中的至少一个点或一条边.确定图的最小基数的混合控制集的问题称为混合控制问题.本文研究混合控制问题的算法复杂性,证明了混合控制问题在无向路图上是NP-完全的,但在块图上有线性时间算法.无向路图和块图都是弦图的子类,又是树的母类.     

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.     

Precoloring Extension of Co-Meyniel Graphs

The pre-coloring extension problem consists, given a graph G and a set of nodes to which some colors are already assigned, in finding a coloring of G with the minimum number of colors which respects the pre-coloring assignment. This can be reduced to the usual coloring problem on a certain contracted graph. We prove that pre-coloring extension is polynomial for complements of Meyniel graphs. We answer a question of Hujter and Tuza by showing that “PrExt perfect” graphs are exactly the co-Meyniel graphs, which also generalizes results of Hujter and Tuza and of Hertz. Moreover we show that, given a co-Meyniel graph, the corresponding contracted graph belongs to a restricted class of perfect graphs (“co-Artemis” graphs, which are “co-perfectly contractile” graphs), whose perfectness is easier to establish than the strong perfect graph theorem. However, the polynomiality of our algorithm still depends on the ellipsoid method for coloring perfect graphs. C.N.R.S. Final version received: January, 2007     

List Homomorphisms and Circular Arc Graphs

G , H, and lists , a list homomorphism of G to H with respect to the lists L is a mapping , such that for all , and for all . The list homomorphism problem for a fixed graph H asks whether or not an input graph G together with lists , , admits a list homomorphism with respect to L. We have introduced the list homomorphism problem in an earlier paper, and proved there that for reflexive graphs H (that is, for graphs H in which every vertex has a loop), the problem is polynomial time solvable if H is an interval graph, and is NP-complete otherwise. Here we consider graphs H without loops, and find that the problem is closely related to circular arc graphs. We show that the list homomorphism problem is polynomial time solvable if the complement of H is a circular arc graph of clique covering number two, and is NP-complete otherwise. For the purposes of the proof we give a new characterization of circular arc graphs of clique covering number two, by the absence of a structure analogous to Gallai's asteroids. Both results point to a surprising similarity between interval graphs and the complements of circular arc graphs of clique covering number two. Received: July 22, 1996/Revised: Revised June 10, 1998     

Star-Uniform Graphs

A star-factor of a graph is a spanning subgraph each of whose components is a star. A graph G is called star-uniform if all star-factors of G have the same number of components. Motivated by the minimum cost spanning tree and the optimal assignment problems, Hartnell and Rall posed an open problem to characterize all the star-uniform graphs. In this paper, we show that a graph G is star-uniform if and only if G has equal domination and matching number. From this point of view, the star-uniform graphs were characterized by Randerath and Volkmann. Unfortunately, their characterization is incomplete. By deploying Gallai–Edmonds Matching Structure Theorem, we give a clear and complete characterization of star-unform graphs.     

Triple Crossing Numbers of Graphs

We introduce the triple crossing number,a variation of the crossing number,of a graph,which is the minimal number of crossing points in all drawings of the graph with only triple crossings.It is defined to be zero for planar graphs,and to be infinite for non-planar graphs which do not admit a drawing with only triple crossings.In this paper,we determine the triple crossing numbers for all complete multipartite graphs which include all complete graphs.     

Coloring Fuzzy Circular Interval Graphs

Computing the weighted coloring number of graphs is a classical topic in combinatorics and graph theory. Recently these problems have again attracted a lot of attention for the class of quasi-line graphs and more specifically fuzzy circular interval graphs.The problem is NP-complete for quasi-line graphs. For the subclass of fuzzy circular interval graphs however, one can compute the weighted coloring number in polynomial time using recent results of Chudnovsky and Ovetsky and of King and Reed. Whether one could actually compute an optimal weighted coloring of a fuzzy circular interval graph in polynomial time however was still open.We provide a combinatorial algorithm that computes weighted colorings and the weighted coloring number for fuzzy circular interval graphs efficiently. The algorithm reduces the problem to the case of circular interval graphs, then making use of an algorithm by Gijswijt to compute integer decompositions.     

双线性型图与交错型图不是完美图

本文证明了双线性型图与交错型图都不是完美图,从而解决了双线性型图与交错型图的完美图判别问题.     

Perfect Matchings of Cellular Graphs

We introduce a family of graphs, called cellular, and consider the problem of enumerating their perfect matchings. We prove that the number of perfect matchings of a cellular graph equals a power of 2 times the number of perfect matchings of a certain subgraph, called the core of the graph. This yields, as a special case, a new proof of the fact that the Aztec diamond graph of order n introduced by Elkies, Kuperberg, Larsen and Propp has exactly 2 ^n(n+1)/2 perfect matchings. As further applications, we prove a recurrence for the number of perfect matchings of certain cellular graphs indexed by partitions, and we enumerate the perfect matchings of two other families of graphs called Aztec rectangles and Aztec triangles.     

The Exact Weighted Independent Set Problem in Perfect Graphs and Related Classes

The exact weighted independent set (EWIS) problem consists in determining whether a given vertex-weighted graph contains an independent set of given weight. This problem is a generalization of two well-known problems, the NP-complete subset sum problem and the strongly NP-hard maximum weight independent set (MWIS) problem. Since the MWIS problem is polynomially solvable for some special graph classes, it is interesting to determine the complexity of this more general EWIS problem for such graph classes.We focus on the class of perfect graphs, which is one of the most general graph classes where the MWIS problem can be solved in polynomial time. It turns out that for certain subclasses of perfect graphs, the EWIS problem is solvable in pseudo-polynomial time, while on some others it remains strongly NP-complete. In particular, we show that the EWIS problem is strongly NP-complete for bipartite graphs of maximum degree three, but solvable in pseudo-polynomial time for cographs, interval graphs and chordal graphs, as well as for some other related graph classes.     

Feasible Graphs and Colorings

The problem of when a recursive graph has a recursive k-coloring has been extensively studied by Bean, Schmerl, Kierstead, Remmel, and others. In this paper, we study the polynomial time analogue of that problem. We develop a number of negative and positive results about colorings of polynomial time graphs. For example, we show that for any recursive graph G and for any k, there is a polynomial time graph G′ whose vertex set is {0,1}* such that there is an effective degree preserving correspondence between the set of k-colorings of G and the set of k-colorings of G′ and hence there are many examples of k-colorable polynomial time graphs with no recursive k-colorings. Moreover, even though every connected 2-colorable recursive graph is recursively 2-colorable, there are connected 2-colorable polynomial time graphs which have no primitive recursive 2-coloring. We also give some sufficient conditions which will guarantee that a polynomial time graph has a polynomial time or exponential time coloring.     

The Inducibility of Graphs on Four Vertices

We consider the problem of determining the maximum induced density of a graph H in any graph on n vertices. The limit of this density as n tends to infinity is called the inducibility of H. The exact value of this quantity is known only for a handful of small graphs and a specific set of complete multipartite graphs. Answering questions of Brown–Sidorenko and Exoo, we determine the inducibility of K_{1, 1, 2} and the paw graph. The proof is obtained using semidefinite programming techniques based on a modern language of extremal graph theory, which we describe in full detail in an accessible setting.     

Orthogonal Polarity Graphs and Sidon Sets

Determining the maximum number of edges in an n‐vertex C₄‐free graph is a well‐studied problem that dates back to a paper of Erd?s from 1938. One of the most important families of C₄‐free graphs are the Erd?s‐Rényi orthogonal polarity graphs. We show that the Cayley sum graph constructed using a Bose‐Chowla Sidon set is isomorphic to a large induced subgraph of the Erd?s‐Rényi orthogonal polarity graph. Using this isomorphism, we prove that the Petersen graph is a subgraph of every sufficiently large Erd?s‐Rényi orthogonal polarity graph.