On a labeling problem in graphs

Motivated by applications in software programming, we consider the problem of covering a graph by a feasible labeling. Given an undirected graph G=(V,E), two positive integers k and t, and an alphabet Σ, a feasible labeling is defined as an assignment of a set L_v⊆Σ to each vertex v∈V, such that (i) |L_v|≤k for all v∈V and (ii) each label α∈Σ is used no more than t times. An edge e={i,j} is said to be covered by a feasible labeling if L_i∩L_j≠0?. G is said to be covered if there exists a feasible labeling that covers each edge e∈E.In general, we show that the problem of deciding whether or not a tree can be covered is strongly NP-complete. For k=2, t=3, we characterize the trees that can be covered and provide a linear time algorithm for solving the decision problem. For fixed t, we present a strongly polynomial algorithm that solves the decision problem; if a tree can be covered, then a corresponding feasible labeling can be obtained in time polynomial in k and the size of the tree. For general graphs, we give a strongly polynomial algorithm to resolve the covering problem for k=2, t=3.     

The reconstruction conjecture for finite simple graphs and associated directed graphs

In this paper, we study the Reconstruction Conjecture for finite simple graphs. Let Γ and ${\mathrm{\Gamma }}^{\prime }$ be finite simple graphs with at least three vertices such that there exists a bijective map $f:V(\mathrm{\Gamma })\to V({\mathrm{\Gamma }}^{\prime })$ and for any $v\in V(\mathrm{\Gamma })$, there exists an isomorphism ${?}_v:\mathrm{\Gamma }?v\to {\mathrm{\Gamma }}^{\prime }?f(v)$. Then we define the associated directed graph $\stackrel{?}{\mathrm{\Gamma }}=\stackrel{?}{\mathrm{\Gamma }}(\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime },f,{\{{?}_v\}}_{v\in V(\mathrm{\Gamma })})$ with two kinds of arrows from the graphs Γ and ${\mathrm{\Gamma }}^{\prime }$, the bijective map f and the isomorphisms ${\{{?}_v\}}_{v\in V(\mathrm{\Gamma })}$. By investigating the associated directed graph $\stackrel{?}{\mathrm{\Gamma }}$, we study when are the two graphs Γ and ${\mathrm{\Gamma }}^{\prime }$ isomorphic.     

The reconstruction problem for certain infinite graphs

We are concerned with the notion of the degree-type (D _G ⁱ )_i∈ω of a graphG, whereD _G ⁱ is defined to be the number of vertices inG with degreei. In the first section the following results are proven:

1. IfG is a connected, locally finite, countably infinite graph such that there exists ani so thatD _G ⁱ andD _G ⁱ⁺¹ are both finite and different from 0, thenG is reconstructible.
2. Locally finite, countably infinite graphsG, for which infinitely manyD _G ⁱ are different from 0 but only finitely manyD _G ⁱ are infinite, are reconstructible.

In the second section we give some results about the reconstructibility of certain locally finite countably infinite interval graphs and show that a reconstruction of a planar, infinite graph has to be planar too.     

A note on coloring sparse random graphs

Coja-Oghlan and Taraz [Amin Coja-Oghlan, Anusch Taraz, Exact and approximative algorithms for coloring , Random Structures and Algorithms 24 (3) (2004) 259-278] presented a graph coloring algorithm that has expected linear running time for random graphs with edge probability p satisfying np≤1.01. In this work, we develop their analysis by exploiting generating function techniques. We show that, in fact, their algorithm colors G_n,p with the minimal number of colors and has expected linear running time, provided that np≤1.33.     

Online coloring graphs with high girth and high odd girth

We give an upper bound for the online chromatic number of graphs with high girth and for graphs with high odd girth generalizing Kierstead’s algorithm for graphs that contain neither a C₃ or C₅ as an induced subgraph.     

Approximating maximum edge 2-coloring in simple graphs

We present a polynomial-time approximation algorithm for legally coloring as many edges of a given simple graph as possible using two colors. It achieves an approximation ratio of roughly 0.842 and runs in O(n³m) time, where n (respectively, m) is the number of vertices (respectively, edges) in the input graph. The previously best ratio achieved by a polynomial-time approximation algorithm was .     

Two methods for the generation of chordal graphs

In this paper two methods for automatic generation of connected chordal graphs are proposed: the first one is based on new results concerning the dynamic maintenance of chordality under edge insertions; the second is based on expansion/merging of maximal cliques. Theoretical and experimental results are presented. In both methods, chordality is preserved along the whole generation process. L. Markenzon’s research is partially supported by grant 301068/2003-8, CNPq, Brazil.     

The Hamiltonian problem on distance-hereditary graphs

In this paper, we first present an O(n+m)-time sequential algorithm to solve the Hamiltonian problem on a distance-hereditary graph G, where n and m are the number of vertices and edges of G, respectively. This algorithm is faster than the previous best known algorithm for the problem which takes O(n²) time. We also give an efficient parallel implementation of our sequential algorithm. Moreover, if G is represented by its decomposition tree form, the problem can be solved optimally in O(logn) time using O((n+m)/logn) processors on an EREW PRAM.     

An asymptotic solution to the cycle decomposition problem for complete graphs

Let m₁,m₂,…,mt be a list of integers. It is shown that there exists an integer N such that for all n?N, the complete graph of order n can be decomposed into edge-disjoint cycles of lengths m₁,m₂,…,mt if and only if n is odd, 3?mi?n for i=1,2,…,t, and . In 1981, Alspach conjectured that this result holds for all n, and that a corresponding result also holds for decompositions of complete graphs of even order into cycles and a perfect matching.     

The extremal function for 3-linked graphs

A graph is k-linked if for every set of 2k distinct vertices {s₁,…,s_k,t₁,…,t_k} there exist disjoint paths P₁,…,P_k such that the endpoints of P_i are s_i and t_i. We prove every 6-connected graph on n vertices with 5n−14 edges is 3-linked. This is optimal, in that there exist 6-connected graphs on n vertices with 5n−15 edges that are not 3-linked for arbitrarily large values of n.     

The rank of random graphs

We show that almost surely the rank of the adjacency matrix of the Erd?s‐Rényi random graph G(n,p) equals the number of nonisolated vertices for any c ln n/n ≤ p ≤ 1/2, where c is an arbitrary positive constant larger than 1/2. In particular, the adjacency matrix of the giant component (a.s.) has full rank in this range. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Solving the path cover problem on circular-arc graphs by using an approximation algorithm

A path cover of a graph G=(V,E) is a family of vertex-disjoint paths that covers all vertices in V. Given a graph G, the path cover problem is to find a path cover of minimum cardinality. This paper presents a simple O(n)-time approximation algorithm for the path cover problem on circular-arc graphs given a set of n arcs with endpoints sorted. The cardinality of the path cover found by the approximation algorithm is at most one more than the optimal one. By using the result, we reduce the path cover problem on circular-arc graphs to the Hamiltonian cycle and Hamiltonian path problems on the same class of graphs in O(n) time. Hence the complexity of the path cover problem on circular-arc graphs is the same as those of the Hamiltonian cycle and Hamiltonian path problems on circular-arc graphs.     

The dominating set problem is fixed parameter tractable for graphs of bounded genus

We describe an algorithm for the dominating set problem with time complexity O((4g+40)^kn²) for graphs of bounded genus g1, where k is the size of the set. It has previously been shown that this problem is fixed parameter tractable for planar graphs. We give a simpler proof for the previous O(8^kn²) result for planar graphs. Our method is a refinement of the earlier techniques.     

Complexity results in graph reconstruction

We investigate the relative complexity of the graph isomorphism problem (GI) and problems related to the reconstruction of a graph from its vertex-deleted or edge-deleted subgraphs (in particular, deck checking (DC) and legitimate deck (LD) problems). We show that these problems are closely related for all amounts c?1 of deletion:

(1)

, , , and .

(2)

For all k?2, and .

(3)

For all k?2, .

(4)

.

(5)

For all k?2, .

For many of these results, even the c=1 case was not previously known.Similar to the definition of reconstruction numbers vrn_∃(G) [F. Harary, M. Plantholt, The graph reconstruction number, J. Graph Theory 9 (1985) 451-454] and ern_∃(G) (see [J. Lauri, R. Scapellato Topics in Graph Automorphism and Reconstruction, London Mathematical Society, Cambridge University Press, Cambridge, 2003, p. 120]), we introduce two new graph parameters, vrn_∀(G) and ern_∀(G), and give an example of a family {G_n}_n?4 of graphs on n vertices for which vrn_∃(G_n)<vrn_∀(G_n). For every k?2 and n?1, we show that there exists a collection of k graphs on (2^k-1+1)n+k vertices with 2ⁿ 1-vertex-preimages, i.e., one has families of graph collections whose number of 1-vertex-preimages is huge relative to the size of the graphs involved.     

Geometric separation and exact solutions for the parameterized independent set problem on disk graphs

We consider the parameterized problem, whether for a given set  of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k non-intersecting disks. For this problem, we expose an algorithm running in time that is—to our knowledge—the first algorithm with running time bounded by an exponential with a sublinear exponent. For λ-precision disk graphs of bounded radius ratio, we show that the problem is fixed parameter tractable with a running time  . The results are based on problem kernelization and a new “geometric ( -separator) theorem” which holds for all disk graphs of bounded radius ratio. The presented algorithm then performs, in a first step, a “geometric problem kernelization” and, in a second step, uses divide-and-conquer based on our new “geometric separator theorem.”     

Maximum induced matching problem on hhd-free graphs

We show that the maximum induced matching problem can be solved on hhd-free graphs in O(m²) time; hhd-free graphs generalize chordal graphs and the previous best bound was O(m³). Then, we consider a technique used by Brandstädt and Hoàng (2008) [4] to solve the problem on chordal graphs. Extending this, we show that for a subclass of hhd-free graphs that is more general than chordal graphs the problem can be solved in linear time. We also present examples to demonstrate the tightness of our results.     

Recognizing graphs without asteroidal triples

We consider the problem of recognizing AT-free graphs. Although there is a simple O(n³) algorithm, no faster method for solving this problem had been known. Here we give three different algorithms which have a better time complexity for graphs which are sparse or have a sparse complement; in particular we give algorithms which recognize AT-free graphs in , , and O(n^2.82+nm). In addition we give a new characterization of graphs with bounded asteroidal number by the help of the knotting graph, a combinatorial structure which was introduced by Gallai for considering comparability graphs.     

The traveling salesman problem in graphs with some excluded minors

Given a graph and a length function defined on its edge-set, the Traveling Salesman Problem can be described as the problem of finding a family of edges (an edge may be chosen several times) which forms a spanning Eulerian subgraph of minimum length. In this paper we characterize those graphs for which the convex hull of all solutions is given by the nonnegativity constraints and the classical cut constraints. This characterization is given in terms of excluded minors. A constructive characterization is also given which uses a small number of basic graphs.     

Lower bounds for the merit factors of trigonometric polynomials from Littlewood classes

With the notation ,

we prove the following result.Theorem 1. Assume that p is a trigonometric polynomial of degree at most n with real coefficients that satisfies

||p||_L2(K)An^1/2 and ||p′||_L2(K)Bn^3/2.

Then

M₄(p)−M₂(p)M₂(p)

with

We also prove that

and

M₂(p)−M₁(p)10⁻³¹M₂(p)

for every , where denotes the collection of all trigonometric polynomials of the form