Subgraph Of Degree Research Articles

Improper colouring of (random) unit disk graphs

For any graph $G$, the $k$-improper chromatic number $χ ^k(G)$ is the smallest number of colours used in a colouring of $G$ such that each colour class induces a subgraph of maximum degree $k$. We investigate the ratio of the $k$-improper chromatic number to the clique number for unit disk graphs and random unit disk graphs to extend results of [McRe99, McD03] (where they considered only proper colouring).

Counting connected graphs inside-out

The theme of this work is an “inside-out” approach to the enumeration of graphs. It is based on a well-known decomposition of a graph into its 2-core, i.e. the largest subgraph of minimum degree 2 or more, and a forest of trees attached. Using our earlier (asymptotic) formulae for the total number of 2-cores with a given number of vertices and edges, we solve the corresponding enumeration problem for the connected 2-cores. For a subrange of the parameters, we also enumerate those 2-cores by using a deeper inside-out notion of a kernel of a connected 2-core. Using this enumeration result in combination with Caley's formula for forests, we obtain an alternative and simpler proof of the asymptotic formula of Bender, Canfield and McKay for the number of connected graphs with n vertices and m edges, with improved error estimate for a range of m values. As another application, we study the limit joint distribution of three parameters of the giant component of a random graph with n vertices in the supercritical phase, when the difference between average vertex degree and 1 far exceeds n - 1 / 3 . The three parameters are defined in terms of the 2-core of the giant component, i.e. its largest subgraph of minimum degree 2 or more. They are the number of vertices in the 2-core, the excess (#edges - #vertices) of the 2-core, and the number of vertices not in the 2-core. We show that the limit distribution is jointly Gaussian throughout the whole supercritical phase. In particular, for the first time, the 2-core size is shown to be asymptotically normal, in the widest possible range of the average vertex degree.

K-Pseudosnakes in n-dimensional Hypercubes

Abstract A k-pseudosnake in a graph is an induced subgraph of maximum degree at most k. In this paper we show that k-pseudosnakes with more than 2 n − 1 vertices exist in the hypercubes Q n , provided n ⩽ 2 k . We also give upper bounds, and show that the generated k-pseudosnakes are maximum provided k is even and n = 3 k / 2 . The results also yield better constructions of k-pseudosnakes in large n-dimensional grids in certain cases.

Towards optimally multiplexed applications of universal arrays.

We study a design and optimization problem that occurs, for example, when single nucleotide polymorphisms (SNPs) are to be genotyped using a universal DNA tag array. The problem of optimizing the universal array to avoid disruptive cross-hybridization between universal components of the system was addressed in previous work. Cross-hybridization can, however, also occur assay specifically, due to unwanted complementarity involving assay-specific components. Here we examine the problem of identifying the most economic experimental configuration of the assay-specific components that avoids cross-hybridization. Our formalization translates this problem into the problem of covering the vertices of one side of a bipartite graph by a minimum number of balanced subgraphs of maximum degree 1. We show that the general problem is NP-complete. However, in the real biological setting, the vertices that need to be covered have degrees bounded by d. We exploit this restriction and develop an O(d)-approximation algorithm for the problem. We also give an O(d)-approximation for a variant of the problem in which the covering subgraphs are required to be vertex disjoint. In addition, we propose a stochastic model for the input data and use it to prove a lower bound on the cover size. We complement our theoretical analysis by implementing two heuristic approaches and testing their performance on synthetic data as well as on simulated SNP data.

Every Graph of Sufficiently Large Average Degree Contains a C 4 -Free Subgraph of Large Average Degree

We prove that for every k there exists d=d(k) such that every graph of average degree at least d contains a subgraph of average degree at least k and girth at least six. This settles a special case of a conjecture of Thomassen.

Algorithms for vertex-partitioning problems on graphs with fixed clique-width

Many vertex-partitioning problems can be expressed within a general framework introduced by Telle and Proskurowski. They showed that optimization problems in this framework can be solved in polynomial time on classes of graphs with bounded tree-width. In this paper, we consider a very similar framework, in relationship with more general classes of graphs: we propose a polynomial time algorithm on classes of graphs with bounded clique-width for all the optimization problems in our framework. These classes of graphs are more general than the classes of graphs with bounded tree-width in the sense that classes of graphs with bounded tree-width have also bounded clique-width (but not necessarily the inverse).Our framework includes problems such as independent (dominating) set, p-dominating set, induced bounded degree subgraph, induced p-regular subgraph, perfect matching cut, graph k-coloring and graph list-k-coloring with cardinality constraints (fixed k). This paper thus provides a second (distinct) framework within which the optimization problems can be solved in polynomial time on classes of graphs with bounded clique-width, after a first framework (called MS1) due to the work of Courcelle, Makowsky and Rotics (for which they obtained a linear time algorithm).

Long Cycles in Graphs on a Fixed Surface

We prove that there exists a function a:N0×R+→N such that (i)If G is a 4-connected graph of order n embedded on a surface of Euler genus g such that the face-width of G is at least a(g, ε), then G can be covered by two cycles each of which has length at least (1−ε)n. We apply this to derive lower bounds for the length of a longest cycle in a graph G on any fixed surface. Specifically, there exist functions b:N0→N and c:N0→R+ such that for every graph G on n vertices that is embedded on a surface of Euler genus g the following statements hold: (ii)If G is 4-connected, then G contains a collection of at most b(g) paths which cover all vertices of G, and G contains a cycle of length at least n/b(g). (iii)If G is 3-connected, then G contains a cycle of length at least c(g)nlog2/log3. Moreover, for each ε>0, every 4-connected graph G with sufficiently large face-width contains a spanning tree of maximum degree at most 3 and a 2-connected spanning subgraph of maximum degree at most 4 such that the number of vertices of degree 3 or 4 in either of these subgraphs is at most ε|V(G)|.

Generalized trees related with tree metrics

Abstract Graphs without induced subgraphs of minimum degree at least 3 have been considered in matroid theory (see, e.g., Welsh 1976, ch. 14 or Aigner 1979, ch. 7) and in some domains of applications (Todd 1989, Guenoche and Leclerc 1999). Maximal such graphs were characterized in Todd's paper. They are called here 2d-trees. A subclass of 2d-trees consists of the classical 2-trees (Pippert and Beineke 1969), whose interesting status among chordal, series parallel and 2-connected graphs has been extensively studied in the literature (Arnborg and proskurowski 1989, Borie, Parker and Tovey 1991, Rose 1974, and others). We consider here valued 2d-trees and 2-trees with a given finite vertex set X.

The parallel complexity of approximating the high degree subgraph problem

The high degree subgraph problem is to find a subgraph H of a graph G such that the minimum degree of H is as large as possible. This problem is known to be P-hard so that parallel approximation algorithms are very important for it. Our first goal is to determine how effectively the approximation algorithm based on a well-known extremal graph result parallelizes. In particular, we show that two natural decision problems associated with this algorithm are P-complete: these results suggest that the parallel implementation of the algorithm itself requires more sophisticated techniques. Successively, we study the high degree subgraph problem for random graphs with any edge probability function and we provide different parallel approximation algorithms depending on the type of this function.

Blow-up Lemma

Regular pairs behave like complete bipartite graphs from the point of view of bounded degree subgraphs.

Regular Subgraphs in Graphs and Rooted Graphs and Definability in Monadic Second ‐ Order Logic

AbstractWe investigate the definability in monadic ∑11 and monadic Π11 of the problems REGk, of whether there is a regular subgraph of degree k in some given graph, and XREGk, of whether, for a given rooted graph, there is a regular subgraph of degree k in which the root has degree k, and their restrictions to graphs in which every vertex has degree at most k, namely REGkk and XREGkk, respectively, for k ≥ 2 (all our graphs are undirected). Our motivation partly stems from the fact (which we prove here) that REGkk and XREGkk are logspace equivalent to CONN and REACH, respectively, for k ≥ 3, where CONN is the problem of whether a given graph is connected and REACH is the problem of whether a given graph has a path joining two given vertices. We use monadic first ‐ order reductions, monadic ∑11 games and a recent technique due to Fagin, Stockmeyer and Vardi to almost completely classify whether these problems are definable in monadic ∑11 and monadic Π11, and we compare the definability of these problems (in monadic ∑11 and monadic Π11 with their computational complexity (which varies from solvable using logspace to NP ‐ complete).

Decomposing graphs under degree constraints

We prove a conjecture of C. Thomassen: If s and t are non-negative integers, and if G is a graph with minimum degree s + t + 1, then the vertex set of G can be partitioned into two sets which induce subgraphs of minimum degree at least s and t, respectively. © 1996 John Wiley & Sons, Inc.

Sizes of graphs with induced subgraphs of large maximum degree

Graphs with n + k vertices in which every set of n + j vertices induce a subgraph of maximum degree at least n are considered. For j = 1 and for k fairly small compared to n, we determine the minimum number of edges in such graphs.

Constructing the highest degree subgraph for dense graphs is in [formula omitted

In this paper, we first show that the Highest Degree Subgraph problem remains P -complete for dense graphs (i.e. when m = Ω( n 2 poly log n ) ). This hardness result gives a clear motivation in studying the approximability of the Highest Degree Problem even for this restricted case. We then provide an N b -approximation scheme computing approximate solutions for dense graphs, thus proving that, in this case, the problem belongs to the N b A S class.

2‐connected coverings of bounded degree in 3‐connected graphs

AbstractIn a recent paper, Barnette showed that every 3‐connected planar graph has a 2‐connected spanning subgraph of maximum degree at most fifteen, he also constructed a planar triangulation that does not have 2‐connected spanning subgraphs of maximum degree five. In this paper, we show that every 3‐connected graph which is embeddable in the sphere, the projective plane, the torus or the Klein bottle has a 2‐connected spanning subgraph of maximum degree at most six. © 1995 John Wiley & Sons, Inc.

Using maximal independent sets to solve problems in parallel

We show that the problem of finding a maximal vertex-induced (resp., edge-induced) subgraph of maximum degree k is in NC 2 for k ⩾ 0 (resp., k ⩾ 1). For these problems, we develop a method which exploits the NC algorithm for the maximal independent set problem. By using the same scheme, the maximal vertex-induced subgraph which satisfies a hereditary graph property π and whose maximum degree is at most Δ can be found in time O( Δ λ( π) T π ( n)( log( n + m)) 2) using a polynomial number of processors, where λ(π) is the maximum of diameters of minimal graphs violating π and T π ( n) is the time needed to decide whether a graph with n vertices and m edges satisfies π.

Long Cycles and 3-Connected Spanning Subgraphs of Bounded Degree in 3-Connected K1, d-Free Graphs

Let G be a 5-connected K 1, d -free graph on n vertices. We show that G contains a 3-connected spanning subgraph of maximum degree at most 2 d − 1. Using an earlier result of ours, we deduce that G contains a cycle of length at least 1 2 n c where c = (log 2 6 + 2 log 2 (2 d − 1)) −1 .

Subgraphs of minimal degree k

For k ⩾ 2, any graph G with n vertices and (k−1)(n−k+2)+(2k−2) edges has a subrgraph of minimum degree at least k; however, this subgraph need not be proper. It is shown that if G has at least (k−1)(n−k+2)+(2k−2)+1 edges, then there is a subgraph H of minimal degree k that has at most n − √n;√6k3vertices. Also, conditions that insure the existense of smaller subgraphs of minimum degree k are given.

Ordered vertex removal and subgraph problems

Several new graph theoretic problems, which arise naturally from existing coloring algorithms, are defined. The complementary High Degree Vertex Removal Problem and Low Degree Vertex Removal Problem are both shown to be NP-complete. The Low Degree Subgraph Problem is defined and shown to be NP-complete, whereas, the “complementary” problem for high degree subgraphs was previously shown to be P-complete. Since the obvious sequential algorithms for computing the low degree subgraph and high degree subgraph are based on high degree vertex removal and low degree vertex removal, respectively, we find this is an interesting result. The “greedy” versions of the vertex removal and subgraph are shown to be P-complete. In addition, a natural lexicographic version of the Low Degree Subgraph Problem is shown to be NP-complete.

Long cycles in graphs with no subgraphs of minimal degree 3

If a graph G has n vertices and 2 n−1 edges, it must contain some proper subgraph of minimal degree 3. If G has one edge fewer and contains no such subgraph, then, as proved by Erdős, Faudree, Gyárfás and Schelp, it contains a cycle of length at least ⦜log n⌊. Our aim in this note is to prove an essentially best possible result, namely that such a graph must contain a cycle of length at least 4 log n+ O(log log n).