The super laceability of the hypercubes

A k-containerC(u,v) of a graph G is a set of k disjoint paths joining u to v. A k-container C(u,v) is a k∗-container if every vertex of G is incident with a path in C(u,v). A bipartite graph G is k∗-laceable if there exists a k∗-container between any two vertices u, v from different partite set of G. A bipartite graph G with connectivity k is super laceable if it is i∗-laceable for all i?k. A bipartite graph G with connectivity k is f-edge fault-tolerant super laceable if G−F is i∗-laceable for any 1?i?k−f and for any edge subset F with |F|=f<k−1. In this paper, we prove that the hypercube graph Q_r is super laceable. Moreover, Q_r is f-edge fault-tolerant super laceable for any f?r−2.     

Composing Power Series Over a Finite Ring in Essentially Linear Time

Fix a finite commutative ringR. Letuandvbe power series overR, withv(0) = 0. This paper presents an algorithm that computes the firstnterms of the compositionu(v), given the firstnterms ofuandv, inn^1 + o(1)ring operations. The algorithm is very fast in practice whenRhas small characteristic.     

Broadcast in the rendezvous model

In many large, distributed or mobile networks, broadcast algorithms are used to update information stored at the nodes. In this paper, we propose a new model of communication based on rendezvous and analyze a multi-hop distributed algorithm to broadcast a message in a synchronous setting. In the rendezvous model, two neighbors u and v can communicate if and only if u calls v and v calls u simultaneously. Thus nodes u and v obtain a rendezvous at a meeting point. If m is the number of meeting points, the network can be modeled by a graph of n vertices and m edges. At each round, every vertex chooses a random neighbor and there is a rendezvous if an edge has been chosen by its two extremities. Rendezvous enable an exchange of information between the two entities. We get sharp lower and upper bounds on the time complexity in terms of number of rounds to broadcast: we show that, for any graph, the expected number of rounds is between ln n and O (n²). For these two bounds, we prove that there exist some graphs for which the expected number of rounds is either O (ln (n)) or Ω (n²). For specific topologies, additional bounds are given.     

On edge connectivity of direct products of graphs

Let λ(G) be the edge connectivity of G. The direct product of graphs G and H is the graph with vertex set V(G×H)=V(G)×V(H), where two vertices (u₁,v₁) and (u₂,v₂) are adjacent in G×H if u₁u₂∈E(G) and v₁v₂∈E(H). We prove that λ(G×Kn)=min{n(n−1)λ(G),(n−1)δ(G)} for every nontrivial graph G and n?3. We also prove that for almost every pair of graphs G and H with n vertices and edge probability p, G×H is k-connected, where k=O(²(n/logn)).     

A polynomial time algorithm for obtaining minimum edge ranking on two-connected outerplanar graphs

An edge ranking of a graph G is a labeling r of its edges with positive integers such that every path between two different edges eu, ev with the same rank r(eu)=r(ev) contains an intermediate edge ew with rank r(ew)>r(eu). An edge ranking of G is minimum if the largest rank k assigned is the smallest among all rankings of G. The edge ranking problem is to find a minimum edge ranking of given graph G. This problem is NP-hard and no polynomial time algorithm for solving it is known for non-trivial classes of graphs other than the class of trees. In this paper, we first show, on a general graph G, a relation between a minimum edge ranking of G and its minimal cuts, which ensures that we can obtain a polynomial time algorithm for obtaining minimum edge ranking of a given graph G if minimal cuts for each subgraph of G can be found in polynomial time and the number of those is polynomial. Based on this relation, we develop a polynomial time algorithm for finding a minimum edge ranking on a 2-connected outerplanar graph.     

The edge-orientation problem and some of its variants on weighted graphs

Let G(V, E) be a connected undirected graph with n vertices and m edges, where each vertex v is associated with a cost C(v) and each edge e = (u, v) is associated with two weights, W(u → v) and W(v → u). The issue of assigning an orientation to each edge so that G becomes a directed graph is resolved in this paper. Determining a scheme to assign orientations of all edges such that max_x∈V{C(x)+∑_x→zW(x→z)} is minimized is the objective. This issue is called the edge-orientation problem (the EOP). Two variants of the EOP, the Out-Degree-EOP and the Vertex-Weighted EOP, are first proposed and then efficient algorithms for solving them on general graphs are designed. Ascertaining that the EOP is NP-hard on bipartite graphs and chordal graphs is the second result. Finally, an O(n log n)-time algorithm for the EOP on trees is designed. In general, the algorithmic results in this paper facilitate the implementation of the weighted fair queuing (WFQ) on real networks. The objective of the WFQ is to assign an effective weight for each flow to enhance link utilization. Our findings consequently can be easily extended to other classes of graphs, such as cactus graphs, block graphs, and interval graphs.     

Geodesic pancyclicity and balanced pancyclicity of Augmented cubes

For two distinct vertices u,v∈V(G), a cycle is called geodesic cycle with u and v if a shortest path of G joining u and v lies on the cycle; and a cycle C is called balanced cycle with u and v if dC(u,v)=max{dC(x,y)|x,y∈V(C)}. A graph G is pancyclic [J. Mitchem, E. Schmeichel, Pancyclic and bipancyclic graphs a survey, Graphs and applications (1982) 271-278] if it contains a cycle of every length from 3 to |V(G)| inclusive. A graph G is called geodesic pancyclic [H.C. Chan, J.M. Chang, Y.L. Wang, S.J. Horng, Geodesic-pancyclic graphs, in: Proceedings of the 23rd Workshop on Combinatorial Mathematics and Computation Theory, 2006, pp. 181-187] (respectively, balanced pancyclic) if for each pair of vertices u,v∈V(G), it contains a geodesic cycle (respectively, balanced cycle) of every integer length of l satisfying max{2dG(u,v),3}?l?|V(G)|. Lai et al. [P.L. Lai, J.W. Hsue, J.J.M. Tan, L.H. Hsu, On the panconnected properties of the Augmented cubes, in: Proceedings of the 2004 International Computer Symposium, 2004, pp. 1249-1251] proved that the n-dimensional Augmented cube, AQn, is pancyclic in the sense that a cycle of length l exists, 3?l?|V(AQn)|. In this paper, we study two new pancyclic properties and show that AQn is geodesic pancyclic and balanced pancyclic for n?2.     

Computing Local Artin Maps,and Solvability of Norm Equations

Let L = K(α) be an Abelian extension of degree n of a number field K, given by the minimal polynomial of α over K. We describe an algorithm for computing the local Artin map associated with the extension L / K at a finite or infinite prime v of K. We apply this algorithm to decide if a nonzero a ∈ K is a norm from L, assuming that L / K is cyclic.     

Approximate Solutions of Polynomial Equations

In this paper, we introduce “approximate solutions" to solve the following problem: given a polynomial F(x, y) over Q, where x represents an n -tuple of variables, can we find all the polynomials G(x) such that F(x, G(x)) is identically equal to a constant c in Q ? We have the following: let F(x, y) be a polynomial over Q and the degree of y in F(x, y) be n. Either there is a unique polynomial g(x)  ∈ Q [ x ], with its constant term equal to 0, such that F(x, y)  = ∑_j = 0ⁿc_j(y − g(x))^jfor some rational numbers c_j, hence, F(x, g(x)  + a)  ∈ Q for all a ∈ Q, or there are at most t distinct polynomials g₁(x),⋯ , g_t(x), t ≤ n, such that F(x, g_i(x))  ∈ Q for 1  ≤ i ≤ t. Suppose that F(x, y) is a polynomial of two variables. The polynomial g(x) for the first case, or g₁(x),⋯ , g_t(x) for the second case, are approximate solutions of F(x, y), respectively. There is also a polynomial time algorithm to find all of these approximate solutions. We then use Kronecker’s substitution to solve the case of F(x, y).     

Adjacent vertex-distinguishing edge and total chromatic numbers of hypercubes

An adjacent vertex-distinguishing edge coloring of a simple graph G is a proper edge coloring of G such that incident edge sets of any two adjacent vertices are assigned different sets of colors. A total coloring of a graph G is a coloring of both the edges and the vertices. A total coloring is proper if no two adjacent or incident elements receive the same color. An adjacent vertex-distinguishing total coloring h of a simple graph G=(V,E) is a proper total coloring of G such that H(u)≠H(v) for any two adjacent vertices u and v, where H(u)={h(wu)|wu∈E(G)}∪{h(u)} and H(v)={h(xv)|xv∈E(G)}∪{h(v)}. The minimum number of colors required for an adjacent vertex-distinguishing edge coloring (resp. an adjacent vertex-distinguishing total coloring) of G is called the adjacent vertex-distinguishing edge chromatic number (resp. adjacent vertex-distinguishing total chromatic number) of G and denoted by (resp. χat(G)). In this paper, we consider the adjacent vertex-distinguishing edge chromatic number and adjacent vertex-distinguishing total chromatic number of the hypercube Qn, prove that for n?3 and χat(Qn)=Δ(Qn)+2 for n?2.     

The super-connected property of recursive circulant graphs

In a graph G, a k-container C_k(u,v) is a set of k disjoint paths joining u and v. A k-container C_k(u,v) is k∗-container if every vertex of G is passed by some path in C_k(u,v). A graph G is k∗-connected if there exists a k∗-container between any two vertices. An m-regular graph G is super-connected if G is k∗-connected for any k with 1?k?m. In this paper, we prove that the recursive circulant graphs G(2^m,4), proposed by Park and Chwa [Theoret. Comput. Sci. 244 (2000) 35-62], are super-connected if and only if m≠2.     

Consensus models: Computational complexity aspects in modern approaches to the list coloring problem

In the paper we study new approaches to the problem of list coloring of graphs. In the problem we are given a simple graph G=(V,E) and, for every v∈V, a nonempty set of integers S(v); we ask if there is a coloring c of G such that c(v)∈S(v) for every v∈V. Modern approaches, connected with applications, change the question—we now ask if S can be changed, using only some elementary transformations, to ensure that there is such a coloring and, if the answer is yes, what is the minimal number of changes. In the paper for studying the adding, the trading and the exchange models of list coloring, we use the following transformations:

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adding of colors (the adding model): select two vertices u, v and a color c∈S(u); add c to S(v), i.e. set S(v):=S(v)∪{c};

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trading of colors (the trading model): select two vertices u, v and a color c∈S(u); move c from S(u) to S(v), i.e. set S(u):=S(u)?{c} and S(v):=S(v)∪{c};

•

exchange of colors (the exchange model): select two vertices u, v and two colors c∈S(u), d∈S(v); exchange c with d, i.e. set S(u):=(S(u)?{c})∪{d} and S(v):=(S(v)?{d})∪{c}.

Our study focuses on computational complexity of the above models and their edge versions. We consider these problems on complete graphs, graphs with bounded cyclicity and partial k-trees, receiving in all cases polynomial algorithms or proofs of NP-hardness.     

Bandwidth on AT-free graphs

We study the classical Bandwidth problem from the viewpoint of parametrised algorithms. Given a graph G=(V,E) and a positive integer k, the Bandwidth problem asks whether there exists a bijective function β:{1,…,∣V∣}→V such that for every edge uv∈E, ∣β⁻¹(u)−β⁻¹(v)∣≤k. It is known that under standard complexity assumptions, no algorithm for Bandwidth with running time of the form f(k)n^O(1) exists, even when the input is restricted to trees. We initiate the search for classes of graphs where such algorithms do exist. We present an algorithm with running time n⋅2^O(klogk) for Bandwidth on AT-free graphs, a well-studied graph class that contains interval, permutation, and cocomparability graphs. Our result is the first non-trivial algorithm that shows fixed-parameter tractability of Bandwidth on a graph class on which the problem remains NP-complete.     

Approximating Node-Connectivity Augmentation Problems

We consider the (undirected) Node Connectivity Augmentation (NCA) problem: given a graph J=(V,E _J ) and connectivity requirements $\{r(u,v): u,v \in V\}$ , find a minimum size set I of new edges (any edge is allowed) such that the graph J∪I contains r(u,v) internally-disjoint uv-paths, for all u,v∈V. In Rooted NCA there is s∈V such that r(u,v)>0 implies u=s or v=s. For large values of k=max? _u,v∈V r(u,v), NCA is at least as hard to approximate as Label-Cover and thus it is unlikely to admit an approximation ratio polylogarithmic in k. Rooted NCA is at least as hard to approximate as Hitting-Set. The previously best approximation ratios for the problem were O(kln?n) for NCA and O(ln?n) for Rooted NCA. In this paper we give an approximation algorithm with ratios O(kln?² k) for NCA and O(ln?² k) for Rooted NCA. This is the first approximation algorithm with ratio independent of?n, and thus is a constant for any fixed k. Our algorithm is based on the following new structural result which is of independent interest. If $\mathcal{D}$ is a set of node pairs in a graph?J, then the maximum degree in the hypergraph formed by the inclusion minimal tight sets separating at least one pair in $\mathcal{D}$ is O(? ²), where ? is the maximum connectivity in J of a pair in $\mathcal{D}$ .     

A study of nonlinear dispersive equations with solitary-wave solutions having compact support

With the use of Adomian decomposition method, the prototypical, genuinely nonlinear K(m,n) equation, u_t+(u^m)_x+(uⁿ)_xxx=0, which exhibits compactons — solitons with finite wavelength — is solved exactly. Two numerical illustrations, K(2,2) and K(3,3), are investigated to illustrate the pertinent features of the proposed scheme. The technique is presented in a general way so that it can be used in nonlinear dispersive equations.     

On the pseudoachromatic number of join of graphs

Given an undirected graph G = (V, E), with vertex set V and edge set E, the pseudoachromatic number ψ(G) of the graph G is the maximum number of colors used to color the vertices in such a way that, for any given pair of colors i, j there exists an edge e = (u, v) ∈ E(G) such that u is colored i and v is colored j. In this paper we give a complete characterization of when the ψ of the join of any two graphs is the sum of the ψ of the two graphs.     

Mason: morphological simplification

The traditional rounding and filleting morphological filters are biased. Hence, as r grows, the rounding R_r (S) of S shrinks and the filleting F_r (S) grows. A shape S is r-regular when R_r (S) = F_r (S) = S. The combinations F_r (R_r (S)) and R_r (F_r (S)) produce nearly r-regular shapes, but retain a bias: F_r (R_r (S)) is usually smaller than S and R_r (F_r (S)) is larger. To overcome this bias, we propose a new filter, called Mason. The r-mortar M_r (S) of S is F_r (S)–R_r (S), and the stability of a point P with respect to S is the smallest value of r for which P belongs to M_r (S). Stability provides important information about the shape’s imbedding that cannot be obtained through traditional topological or differential analysis tools. F_r (R_r (S)) and R_r (F_r (S)) only affect space in M_r (S). For each maximally connected component of M_r (S), Mason performs either F_r (R_r (S)) or R_r (F_r (S)), choosing the combination that alters the smallest portion of that component. Hence, Mason acts symmetrically on the shape and on its complement. Its output is guaranteed to have a smaller symmetric difference with the original shape than that of either combination F_r (R_r (S)) or R_r (F_r (S)). Many previously proposed shape simplification algorithms were focused on reducing the combinatorial storage or processing costs of a shape at the expense of the smoothness and regularity or altered the shape in regular portions that did not exhibit any high frequency complexity. Mason is the first shape simplification operator that is independent of the particular representation and offers the advantage of preserving portions of the boundary of S that are regular at the desired scale.     

Approximate Shortest Paths Avoiding a Failed Vertex: Near Optimal Data Structures for Undirected Unweighted Graphs

Let G=(V,E) be an undirected unweighted graph. A path between any two vertices u,v∈V is said to be t-approximate shortest path if its length is at most t times the length of the shortest path between u and v. We address the problem of building a compact data structure which can efficiently answer the following query for any u,v,x∈V and t>1: Report t-approximate shortest path between u and v when vertex x fails. We present data structures for the single source as well as all-pairs versions of this problem. The query time guaranteed by our data structures is optimal up to a constant factor. Moreover, the size of each of them nearly matches the size of the corresponding data structure with no failures.     

An Efficient Algorithm for the k-Pairwise Disjoint Paths Problem in Hypercubes

A graph G(V, E) (|V|⩾2k) satisfies property A_k if, given k pairs of distinct nodes (s₁, t₁), …, (s_k, t_k) of V(G), there are k mutually node-disjoint paths, one connecting s_i and t_i for each i, 1⩽i⩽k. A necessary condition for any graph to satisfy A_k is that it is (2k−1)-connected. Hypercubes are important interconnection topologies for parallel computation and communication networks. It has been known that hypercubes of dimension n (which are n-connected) satisfy A_⌈n/2⌉. In this paper we give an algorithm which, given k=⌈n/2⌉ pairs of distinct nodes (s₁, t₁), …, (s_k, t_k) in the n-dimensional hypercube, finds the k disjoint paths of length at most n+⌈log n⌉+1 in O(n² log* n) time.     

Signed and minus clique-transversal functions on graphs

A minus (respectively, signed) clique-transversal function of a graph G=(V,E) is a function (respectively, {−1,1}) such that _∑u∈Cf(u)?1 for every maximal clique C of G. The weight of a minus (respectively, signed) clique-transversal function of G is f(V)=_∑v∈Vf(v). The minus (respectively, signed) clique-transversal problem is to find a minus (respectively, signed) clique-transversal function of G of minimum weight. In this paper, we present a unified approach to these two problems on strongly chordal graphs. Notice that trees, block graphs, interval graphs, and directed path graphs are subclasses of strongly chordal graphs. We also prove that the signed clique-transversal problem is NP-complete for chordal graphs and planar graphs.