Strongly chordal and chordal bipartite graphs are sandwich monotone

A graph class is sandwich monotone if, for every pair of its graphs G ₁=(V,E ₁) and G ₂=(V,E ₂) with E ₁⊂E ₂, there is an ordering e ₁,…,e _k of the edges in E ₂∖E ₁ such that G=(V,E ₁∪{e ₁,…,e _i }) belongs to the class for every i between 1 and k. In this paper we show that strongly chordal graphs and chordal bipartite graphs are sandwich monotone, answering an open question by Bakonyi and Bono (Czechoslov. Math. J. 46:577–583, 1997). So far, very few classes have been proved to be sandwich monotone, and the most famous of these are chordal graphs. Sandwich monotonicity of a graph class implies that minimal completions of arbitrary graphs into that class can be recognized and computed in polynomial time. For minimal completions into strongly chordal or chordal bipartite graphs no polynomial-time algorithm has been known. With our results such algorithms follow for both classes. In addition, from our results it follows that all strongly chordal graphs and all chordal bipartite graphs with edge constraints can be listed efficiently.     

On cubic 2-independent Hamiltonian connected graphs

A graph G=(V,E) is Hamiltonian connected if there exists a Hamiltonian path between any two vertices in G. Let P ₁=(u ₁,u ₂,…,u _|V|) and P ₂=(v ₁,v ₂,…,v _|V|) be any two Hamiltonian paths of G. We say that P ₁ and P ₂ are independent if u ₁=v ₁,u _|V|=v _|V|, and u _i ≠v _i for 1<i<|V|. A cubic graph G is 2-independent Hamiltonian connected if there are two independent Hamiltonian paths between any two different vertices of G. A graph G is 1-Hamiltonian if G−F is Hamiltonian for any F⊆V∪E with |F|=1. A graph G is super 3^*-connected if there exist i internal disjoint paths spanning G for i=1,2,3. It is proved that every super 3^*-connected graph is 1-Hamiltonian. In this paper, we prove that every cubic 2-independent Hamiltonian connected graph is also 1-Hamiltonian. We present some cubic 2-independent Hamiltonian connected graphs that are super 3^*-connected, some cubic 2-independent Hamiltonian connected graphs that are not super 3^*-connected, some super 3^*-connected graphs that are not 2-independent Hamiltonian connected, and some cubic 1-Hamiltonian graphs that are Hamiltonian connected but neither 2-independent Hamiltonian connected nor super 3^*-connected. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. This work was supported in part by the National Science Council of the Republic of China under Contract NSC 94-2213-E-233-011.     

Linear time construction of 5-phylogenetic roots for tree chordal graphs

Inspired by phylogenetic tree construction in computational biology, Lin et al. (The 11th Annual International Symposium on Algorithms and Computation (ISAAC 2000), pp. 539–551, 2000) introduced the notion of a k -phylogenetic root. A k-phylogenetic root of a graph G is a tree T such that the leaves of T are the vertices of G, two vertices are adjacent in G precisely if they are within distance k in T, and all non-leaf vertices of T have degree at least three. The k-phylogenetic root problem is to decide whether such a tree T exists for a given graph G. In addition to introducing this problem, Lin et al. designed linear time constructive algorithms for k≤4, while left the problem open for k≥5. In this paper, we partially fill this hole by giving a linear time constructive algorithm to decide whether a given tree chordal graph has a 5-phylogenetic root; this is the largest class of graphs known to have such a construction.     

Boundary graph classes for some maximum induced subgraph problems

The notion of a boundary graph class was recently introduced for a classification of hereditary graph classes according to the complexity of a considered problem. Two concrete graph classes are known to be boundary for several graph problems. We formulate a criterion to determine whether these classes are boundary for a given graph problem or not. We also demonstrate that the classes are simultaneously boundary for some continuous set of graph problems and they are not simultaneously boundary for another set of the same cardinality. Both families of problems are constituted by variants of the maximum induced subgraph problem.     

Finding the anti-block vital edge of a shortest path between two nodes

Let P _G (s,t) denote a shortest path between two nodes s and t in an undirected graph G with nonnegative edge weights. A detour at a node u∈P _G (s,t)=(s,…,u,v,…,t) is defined as a shortest path P _G−e (u,t) from u to t which does not make use of (u,v). In this paper, we focus on the problem of finding an edge e=(u,v)∈P _G (s,t) whose removal produces a detour at node u such that the ratio of the length of P _G−e (u,t) to the length of P _G (u,t) is maximum. We define such an edge as an anti-block vital edge (AVE for short), and show that this problem can be solved in O(mn) time, where n and m denote the number of nodes and edges in the graph, respectively. Some applications of the AVE for two special traffic networks are shown. This research is supported by NSF of China under Grants 70471035, 70525004, 701210001 and 60736027, and PSF of China under Grant 20060401003.     

A fast exact algorithm for the problem of optimum cooperation and the structure of its solutions

Given a graph G=(V,E) with edge weights w _e ∈ℝ, the optimum cooperation problem consists in determining a partition of the graph that maximizes the sum of weights of the edges with nodes in the same class plus the number of the classes of the partition. The problem is also known in the literature as the optimum attack problem in networks. Furthermore, a relevant physics application exists.     

A parameterized perspective on packing paths of length two

We study (vertex-disjoint) packings of paths of length two (i.e., of P ₂’s) in graphs under a parameterized perspective. Starting from a maximal P ₂-packing ℘ of size j we use extremal combinatorial arguments for determining how many vertices of ℘ appear in some P ₂-packing of size (j+1) (if such a packing exists). We prove that one can ‘reuse’ 2.5j vertices. We also show that this bound is asymptotically sharp. Based on a WIN-WIN approach, we build an algorithm which decides, given a graph, if a P ₂-packing of size at least k exists in time O^*(2.448^3k)\mathcal{O}^{*}(2.448^{3k}) .     

On the equitable k *-laceability of hypercubes

Let G be a finite undirected bipartite graph. Let u, v be two vertices of G from different partite sets. A collection of k internal vertex disjoint paths joining u to v is referred as a k-container C _k (u,v). A k-container is a k ^*-container if it spans all vertices of G. We define G to be a k ^*-laceable graph if there is a k ^*-container joining any two vertices from different partite sets. A k ^*-container C _k ^*(u,v)={P ₁,…,P _k } is equitable if ||V(P _i )|−|V(P _j )||≤2 for all 1≤i,j≤k. A graph is equitably k ^*-laceable if there is an equitable k ^*-container joining any two vertices in different partite sets. Let Q _n be the n-dimensional hypercube. In this paper, we prove that the hypercube Q _n is equitably k ^*-laceable for all k≤n−4 and n≥5. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. The work of H.-M. Huang was supported in part by the National Science Council of the Republic of China under NSC94-2115-M008-013.     

Compact labelings for efficient first-order model-checking

We consider graph properties that can be checked from labels, i.e., bit sequences, of logarithmic length attached to vertices. We prove that there exists such a labeling for checking a first-order formula with free set variables in the graphs of every class that is nicely locally clique-width-decomposable. This notion generalizes that of a nicely locally tree-decomposable class. The graphs of such classes can be covered by graphs of bounded clique-width with limited overlaps. We also consider such labelings for bounded first-order formulas on graph classes of bounded expansion. Some of these results are extended to counting queries.     

The optimum traversal of a graph

For a given graph (network) having costs [c_ij] associated with its links, the present paper examines the problem of finding a cycle which traverses every link of the graph at least once, and which incurs the minimum cost of traversal. This problem (called thegraph traversal problem, or theChinese postman problem [9]) can be formulated in ways analogous to those used for the well-known travelling salesman problem, and using this apparent similarity, Bellman and Cooke [1] have produced a dynamic programming formulation. This method of solution of the graph traversal problem requires computational times which increase exponentially with the number of links in the graph. Approximately the same rate of increase of computational effort with problem size would result by any other method adapting a travelling salesman algorithm to the present problem.This paper describes an efficient algorithm for the optimal solution of the graph traversal problem based on the matching method of Edmonds [5, 6]. The computational time requirements of this algorithm increase as a low order (2 or 3) power of the number of links in the graph. Computational results are given for graphs of up to 50 vertices and 125 links.The paper then discusses a generalised version of the graph traversal problem, where not one but a number of cycles are required to traverse the graph. In this case each link has (in addition to its cost) a quantity q_ij associated with it, and the sum of the quantities of the links in any one cycle must be less than a given amount representing the cycle capacity. A heuristic algorithm for the solution of this problem is given. The algorithm is based on the optimal algorithm for the single-cycle graph traversal problem and is shown to produce near-optimal results.There is a large number of possible applications where graph traversal problems arise. These applications include: the spraying of roads with salt-grit to prevent ice formation, the inspection of electric power lines, gas, or oil pipelines for faults, the delivery of letter post, etc.     

Structure of Fibonacci cubes: a survey

The Fibonacci cube Γ _n is the subgraph of the n-cube induced by the binary strings that contain no two consecutive 1s. These graphs are applicable as interconnection networks and in theoretical chemistry, and lead to the Fibonacci dimension of a graph. In this paper a survey on Fibonacci cubes is given with an emphasis on their structure, including representations, recursive construction, hamiltonicity, degree sequence and other enumeration results. Their median nature that leads to a fast recognition algorithm is discussed. The Fibonacci dimension of a graph, studies of graph invariants on Fibonacci cubes, and related classes of graphs are also presented. Along the way some new short proofs are given.     

On the distance paired domination of generalized Petersen graphs <Emphasis Type="Italic">P</Emphasis>(<Emphasis Type="Italic">n</Emphasis>,1) and <Emphasis Type="Italic">P</Emphasis>(<Emphasis Type="Italic">n</Emphasis>,2)

Let G=(V,E) be a graph without an isolated vertex. A set D⊆V(G) is a k -distance paired dominating set of G if D is a k-distance dominating set of G and the induced subgraph 〈D〉 has a perfect matching. The minimum cardinality of a k-distance paired dominating set for graph G is the k -distance paired domination number, denoted by γ _p ^k (G). In this paper, we determine the exact k-distance paired domination number of generalized Petersen graphs P(n,1) and P(n,2) for all k≥1.     

On Split-Coloring Problems

We study a new coloring concept which generalizes the classical vertex coloring problem in a graph by extending the notion of stable sets to split graphs. First of all, we propose the packing problem of finding the split graph of maximum size where a split graph is a graph G = (V,E) in which the vertex set V can be partitioned into a clique K and a stable set S. No condition is imposed on the edges linking vertices in S to the vertices in K. This maximum split graph problem gives rise to an associated partitioning problem that we call the split-coloring problem. Given a graph, the objective is to cover all his vertices by a least number of split graphs. Definitions related to this new problem are introduced. We mention some polynomially solvable cases and describe open questions on this area. An erratum to this article is available at .     

On consecutive edge magic total labelings of connected bipartite graphs

Since Sedlá\(\breve{\hbox {c}}\)ek introduced the notion of magic labeling of a graph in 1963, a variety of magic labelings of a graph have been defined and studied. In this paper, we study consecutive edge magic labelings of a connected bipartite graph. We make a useful observation that there are only four possible values of b for which a connected bipartite graph has a b-edge consecutive magic labeling. On the basis of this fundamental result, we deduce various interesting results on consecutive edge magic labelings of bipartite graphs. As a matter of fact, we do not focus just on specific classes of graphs, but also discuss the more general classes of non-bipartite and bipartite graphs.     

带机会约束的动态投资决策模型研究

本文在BlackScholes型市场中,建立了具有投资机会约束的CaR动态投资决策模型: ,其中x是初始财富,π(t)=(π₁(t),…,π_d(t))′∈R^d为可行的证券组合过程,X^π(T)为计划期末的财富水平,CaR(x,π,T)为投资期末的在险资本,R是投资者事先给定的某正的财富水平,0<β<1通过对该模型的讨论,得到了最优常数再调整策略的显式表达式,其金融学含义包括:对于机会约束下的动态投资组合,在风险中性市场中,最优的常数再调整投资策略是纯债券投资策略,最优的在险资本值为零;在风险非中性市场中,最优的常数再调整投资策略蕴涵了共同基金定理的成立。     

Improving an upper bound on the size of k-regular induced subgraphs

This paper considers the NP-hard graph problem of determining a maximum cardinality subset of vertices inducing a k-regular subgraph. For any graph G, this maximum will be denoted by α _k (G). From a well known Motzkin-Straus result, a relationship is deduced between α _k (G) and the independence number α(G). Next, it is proved that the upper bounds υ _k (G) introduced in Cardoso et al. (J. Comb. Optim., 14, 455–463, 2007) can easily be computed from υ ₀(G), for any positive integer k. This relationship also allows one to present an alternative proof of the Hoffman bound extension introduced in the above paper. The paper continues with the introduction of a new upper bound on α _k (G) improving υ _k (G). Due to the difficulty of computing this improved bound, two methods are provided for approximating it. Finally, some computational experiments which were performed to compare all bounds studied are reported.     

L(2,1)-labelings of the edge-path-replacement of a graph

For two positive integers j and k with j≥k, an L(j,k)-labeling of a graph G is an assignment of nonnegative integers to V(G) such that the difference between labels of adjacent vertices is at least j, and the difference between labels of vertices that are distance two apart is at least k. The span of an L(j,k)-labeling of a graph G is the difference between the maximum and minimum integers used by it. The L(j,k)-labelings-number of G is the minimum span over all L(j,k)-labelings of G. This paper focuses on L(2,1)-labelings-number of the edge-path-replacement G(P _k ) of a graph G. Note that G(P ₃) is the incidence graph of G. L(2,1)-labelings of the edge-path-replacement G(P ₃) of a graph, called (2,1)-total labeling of G, was introduced by Havet and Yu in 2002 (Workshop graphs and algorithms, Dijon, France, 2003; Discrete Math. 308:498–513, 2008). They (Havet and Yu, Discrete Math. 308:498–513, 2008) obtain the bound $\Delta+1\leq\lambda^{T}_{2}(G)\leq2\Delta+1$ and conjectured $\lambda^{T}_{2}(G)\leq\Delta+3$ . In this paper, we obtain that λ(G(P _k ))≤Δ+2 for k≥5, and conjecture λ(G(P ₄))≤Δ+2 for any graph G with maximum degree Δ.     

b-Tree Facets for the Simple Graph Partitioning Polytope

The simple graph partitioning problem is to partition an edge-weighted graph into mutually disjoint subgraphs, each consisting of no more than b nodes, such that the sum of the weights of all edges in the subgraphs is maximal. In this paper we introduce a large class of facet defining inequalities for the simple graph partitioning polytopes _n (b), b 3, associated with the complete graph on n nodes. These inequalities are induced by a graph configuration which is built upon trees of cardinality b. We provide a closed-form theorem that states all necessary and sufficient conditions for the facet defining property of the inequalities.     

Robot Map Verification of a Graph World

In the map verification problem, a robot is given a (possibly incorrect) map M of the world G with its position and orientation indicated on the map. The task is to find out whether this map, for the given robot position and its orientation in the map, is correct for the world G. We consider the world model of a graph G = (V _G, E _G) in which, for each vertex, edges incident to the vertex are ordered cyclically around that vertex. (This also holds for the map M = (V _M, E _M.) The robot can traverse edges and enumerate edges incident on the current vertex, but it cannot distinguish vertices (and edges) from each other. To solve the verification problem, the robot uses a portable edge marker, that it can put down at an edge of the graph world G and pick up later as needed. The robot can recognize the edge marker when it encounters it in the world G. By reducing the verification problem to an exploration problem, verification can be completed in O(|V _G| × |E _G|) edge traversals (the mechanical cost) with the help of a single vertex marker which can be dropped and picked up at vertices of the graph world (G. Dudek, M. Jenkin, E. Milios, and D. Wilkes, IEEE Trans. on Robotics and Automation, vol. 7, pp. 859–865, 1991; Robotics and Autonomous Systems, vol. 22(2), pp. 159–178, 1997). In this paper, we show a strategy that verifies a map in O(|V _M|) edge traversals only, using a single edge marker, when M is a plane embedded graph, even though G may not be planar (e.g., G may contain overpasses, tunnels, etc.).     

L(d,1)-labelings of the edge-path-replacement of a graph

For two positive integers j and k with j≥k, an L(j,k)-labeling of a graph G is an assignment of nonnegative integers to V(G) such that the difference between labels of adjacent vertices is at least j, and the difference between labels of vertices that are distance two apart is at least k. The span of an L(j,k)-labeling of a graph G is the difference between the maximum and minimum integers used by it. The L(j,k)-labelings-number of G is the minimum span over all L(j,k)-labelings of G. This paper focuses on L(d,1)-labelings-number of the edge-path-replacement G(P _k ) of a graph G. Note that G(P ₃) is the incidence graph of G. L(d,1)-labelings of the edge-path-replacement G(P _k ) of a graph, called (d,1)-total labeling of G, was introduced in 2002 by Havet and Yu (Workshop graphs and algorithms, 2003; Discrete Math 308:493–513, 2008). Havet and Yu (Discrete Math 308:498–513, 2008) obtained the bound $\Delta+ d-1\leq\lambda^{T}_{d}(G)\leq2\Delta+ d-1$ and conjectured $\lambda^{T}_{d}(G)\leq\Delta+2d-1$ . In (Lü in J Comb Optim, to appear; Zhejiang University, submitted), we worked on L(2,1)-labelings-number and L(1,1)-labelings-number of the edge-path-replacement G(P _k ) of a graph G, and obtained that λ(G(P _k ))≤Δ+2 for k≥5, and conjecture λ(G(P ₄))≤Δ+2 for any graph G with maximum degree Δ. In this paper, we will study L(d,1)-labelings-number of the edge-path-replacement G(P _k ) of a graph G for d≥3 and k≥4.