Degenerate Graphs Research Articles

Efficient enumeration of dominating sets for sparse graphs

A dominating set D of a graph G is a set of vertices such that any vertex in G is in D or its neighbor is in D. Enumeration of minimal dominating sets in a graph is one of the central problems in enumeration study since enumeration of minimal dominating sets corresponds to the enumeration of minimal hypergraph transversals. The output-polynomial time enumeration of minimal hypergraph transversals is an interesting open problem. On the other hand, enumeration of dominating sets including non-minimal ones has not been received much attention. In this paper, we address enumeration problems for dominating sets from sparse graphs which are degenerate graphs and graphs with large girth, and we propose two algorithms for solving the problems. The first algorithm enumerates all the dominating sets for a k-degenerate graph in Ok time per solution using On+m space, where n and m are respectively the number of vertices and edges in an input graph. That is, the algorithm is optimal for graphs with constant degeneracy such as trees, planar graphs, H-minor free graphs with some fixed H. The second algorithm enumerates all the dominating sets in constant time per solution for input graphs with girth at least nine.

Circular Chromatic Number of Signed Graphs

A signed graph is a pair $(G, \sigma)$, where $G$ is a graph (loops and multi edges allowed) and $\sigma: E(G) \to \{+, -\}$ is a signature which assigns to each edge of $G$ a sign. Various notions of coloring of signed graphs have been studied. In this paper, we extend circular coloring of graphs to signed graphs. Given a signed graph $(G, \sigma)$ with no positive loop, a circular $r$-coloring of $(G, \sigma)$ is an assignment $\psi$ of points of a circle of circumference $r$ to the vertices of $G$ such that for every edge $e=uv$ of $G$, if $\sigma(e)=+$, then $\psi(u)$ and $\psi(v)$ have distance at least $1$, and if $\sigma(e)=-$, then $\psi(v)$ and the antipodal of $\psi(u)$ have distance at least $1$. The circular chromatic number $\chi_c(G, \sigma)$ of a signed graph $(G, \sigma)$ is the infimum of those $r$ for which $(G, \sigma)$ admits a circular $r$-coloring. For a graph $G$, we define the signed circular chromatic number of $G$ to be $\max\{\chi_c(G, \sigma): \sigma \text{ is a signature of $G$}\}$. 
 We study basic properties of circular coloring of signed graphs and develop tools for calculating $\chi_c(G, \sigma)$. We explore the relation between the circular chromatic number and the signed circular chromatic number of graphs, and present bounds for the signed circular chromatic number of some families of graphs. In particular, we determine the supremum of the signed circular chromatic number of $k$-chromatic graphs of large girth, of simple bipartite planar graphs, $d$-degenerate graphs, simple outerplanar graphs and series-parallel graphs. We construct a signed planar simple graph whose circular chromatic number is $4+\frac{2}{3}$. This is based and improves on a signed graph built by Kardos and Narboni as a counterexample to a conjecture of Máčajová, Raspaud, and Škoviera.

Recognizing graphs close to bipartite graphs with an application to colouring reconfiguration

AbstractWe continue research into a well‐studied family of problems that ask whether the vertices of a given graph can be partitioned into sets and , where is an independent set and induces a graph from some specified graph class . We consider the case where is the class of ‐degenerate graphs. This problem is known to be polynomial‐time solvable if (recognition of bipartite graphs), but ‐complete if (near‐bipartite graphs) even for graphs of maximum degree 4. Yang and Yuan showed that the case is polynomial‐time solvable for graphs of maximum degree 3. This also follows from a result of Catlin and Lai. We study the general case for ‐vertex graphs of maximum degree . We show how to find and in time for , and in time for . Together, these results provide an algorithmic version of a result of Catlin and also provide an algorithmic version of a generalization of Brook's Theorem, proved by Borodin et al. and Matamala. The results also enable us to solve an open problem of Feghali et al. For a given graph and positive integer , the vertex colouring reconfiguration graph of has as its vertex set the set of ‐colourings of and contains an edge between each pair of colourings that differ on exactly on vertex. We complete the complexity classification of the problem of finding a path in the reconfiguration graph between two given ‐colourings of a given graph of maximum degree .

A constant amortized time enumeration algorithm for independent sets in graphs with bounded clique number

In this study, we address the independent set enumeration problem. Although several efficient enumeration algorithms and careful analyses have been proposed for maximal independent sets, no fine-grained analysis has been given for the non-maximal variant. As the main result, we propose an enumeration algorithm for the non-maximal variant that runs in O(q) amortized time and linear space, where q is the clique number, i.e., the maximum size of a clique in an input graph. Note that the proposed algorithm works correctly even if the exact value of q is unknown. It is optimal for graphs with a bounded clique number, such as, triangle-free graphs, bipartite graphs, planar graphs, bounded degenerate graphs, nowhere dense graphs, and F-free graphs for any fixed graph F, where a F-free graph is a graph that has no copy of F as a subgraph. Furthermore, with a slight modification of our proposed algorithm, we can enumerate independent sets with the size at most k in the same time and space complexity. This problem is a generalization of the original problem since this is equal to the original problem if k=n.

Theoretical aspects of equitable partition of networks into sparse modules

The problem of partitioning a large complex network equitably into sparse modules with given rules can be modeled by the equitable list d-degenerate coloring of graphs. This paper establishes theoretical results on such a coloring based on a newly proposed conjecture which states that every graph G is equitably d-degenerate k-colorable and equitably d-degenerate k-choosable for every integer k≥(Δ(G)+1)/(d+1). This conjecture is strong as it implies the Hajnal-Szemerédi theorem on equitable coloring, the equitable list coloring conjecture (Kostochka, Pelsmajer, and West, 2003), the equitable vertex arboricity conjecture (Wu, Zhang, and Li, 2013), and the equitable list vertex arboricity conjecture (Zhang, 2016). In this paper, we confirm this unified conjecture for globally coupled networks, (d+1)-degenerate graphs, graphs with bounded maximum average degree, and planar graphs with large maximum degree. The equitable d-degenerate k-colorability part of this conjecture is also verified for interval graphs, generalizing a result of Niu, Li, and Zhang (2021).

Hat Guessing Numbers of Degenerate Graphs

Recently, Farnik asked whether the hat guessing number $\mathrm{HG}(G)$ of a graph $G$ could be bounded as a function of its degeneracy $d$, and Bosek, Dudek, Farnik, Grytczuk and Mazur showed that $\mathrm{HG}(G)\ge 2^d$ is possible. We show that for all $d\ge 1$ there exists a $d$-degenerate graph $G$ for which $\mathrm{HG}(G) \ge 2^{2^{d-1}}$. We also give a new general method for obtaining upper bounds on $\mathrm{HG}(G)$. The question of whether $\mathrm{HG}(G)$ is bounded as a function of $d$ remains open.

Graph operations and neighborhood polynomials

The neighborhood polynomial of graph $G$ is the generating function for the number of vertex subsets of $G$ of which the vertices have a common neighbor in $G$. In this paper, we investigate the behavior of this polynomial under several graph operations. Specifically, we provide an explicit formula for the neighborhood polynomial of the graph obtained from a given graph $G$ by vertex attachment. We use this result to propose a recursive algorithm for the calculation of the neighborhood polynomial. Finally, we prove that the neighborhood polynomial can be found in polynomial-time in the class of $k$-degenerate graphs.

Decomposing Degenerate Graphs into Locally Irregular Subgraphs

A (undirected) graph is locally irregular if no two of its adjacent vertices have the same degree. A decomposition of a graph G into k locally irregular subgraphs is a partition E_1,...,E_k of E(G) into k parts each of which induces a locally irregular subgraph. Not all graphs decompose into locally irregular subgraphs; however, it was conjectured that, whenever a graph does, it should admit such a decomposition into at most three locally irregular subgraphs. This conjecture was verified for a few graph classes in recent years. This work is dedicated to the decomposability of degenerate graphs with low degeneracy. Our main result is that decomposable k-degenerate graphs decompose into at most 3k+1 locally irregular subgraphs, which improves on previous results whenever k≤9. We improve this result further for some specific classes of degenerate graphs, such as bipartite cacti, k-trees, and planar graphs. Although our results provide only little progress towards the leading conjecture above, the main contribution of this work is rather the decomposition schemes and methods we introduce to prove these results.

Toward Cereceda's conjecture for planar graphs

AbstractThe reconfiguration graph of the ‐colorings of a graph has as vertex set the set of all possible ‐colorings of and two colorings are adjacent if they differ on the color of exactly one vertex. Cereceda conjectured 10 years ago that, for every ‐degenerate graph on vertices, has diameter . The conjecture is wide open, with a best known bound of , even for planar graphs. We improve this bound for planar graphs to . Our proof can be transformed into an algorithm that runs in time.

Fractional Coloring Methods with Applications to Degenerate Graphs and Graphs on Surfaces

We study methods for finding strict upper bounds on the fractional chromatic number $\chi_f(G)$ of a graph $G$. We illustrate these methods by providing short proofs of known inequalities in connection with Grötzsch's 3-color theorem and the 5-color theorem for planar graphs. We also apply it to $d$-degenerate graphs and conclude that every $K_{d+1}$-free $d$-degenerate graph with $n$ vertices has independence number $<n/(d+1)$. We show that for each surface $S$ and every $\varepsilon>0$, the fractional chromatic number of any graph embedded on $S$ of sufficiently large width (depending only on $S$ and $\varepsilon$) is at most $4+\varepsilon$. In the same spirit we prove that Eulerian triangulations or triangle-free graphs of large width have $\chi_f\le 3+\varepsilon$, and quadrangulations of large width have $\chi_f\le 2+\varepsilon$. While the $\varepsilon$ is needed in the latter two results, we conjecture that in the first result $4+\varepsilon$ can be replaced by 4. The upper bounds $\chi_f\le 4+\varepsilon$, $\chi_f\le 3+\varepsilon$, $\chi_f\le 2+\varepsilon$, respectively, are already known for graphs on orientable surfaces, but our results are also valid for graphs on nonorientable surfaces. Surprisingly, a strict lower bound on the fractional chromatic number may imply an upper bound on the chromatic number: Grötzsch's theorem implies that every 4-chromatic planar graph $G$ has fractional chromatic number $\chi_f(G)\ge 3$. We conjecture that this inequality is always strict and observe that this implies the 4-color theorem for planar graphs.

Lossy Kernels for Connected Dominating Set on Sparse Graphs

For $\alpha > 1$, an $\alpha$-approximate (bi)kernel is a polynomial-time algorithm that takes as input an instance $(I, k)$ of a problem $\mathcal{Q}$ and outputs an instance $(I',k')$ (of a problem $\mathcal{Q}'$) of size bounded by a function of $k$ such that, for every $c\geq 1$, a $c$-approximate solution for the new instance can be turned into a $(c\cdot\alpha)$-approximate solution of the original instance in polynomial time. This framework of lossy kernelization was recently introduced by Lokshtanov and co-authors. We study Connected Dominating Set (and its distance-$r$ variant) parameterized by solution size on sparse graph classes like biclique-free graphs, classes of bounded expansion, and nowhere dense classes. We prove that for every $\alpha>1$, Connected Dominating Set admits a polynomial-size $\alpha$-approximate (bi)kernel on all the aforementioned classes. Our results are in sharp contrast to the kernelization complexity of Connected Dominating Set, which is known to not admit a polynomial kernel even on $2$-degenerate graphs and graphs of bounded expansion, unless ${NP} \subseteq \textsf{coNP/poly}$. We complement our results by the following conditional lower bound. We show that if a class $\mathcal{C}$ is somewhere dense and closed under taking subgraphs, then for some value of $r\in \mathbb{N}$ there cannot exist an $\alpha$-approximate bi-kernel for the (Connected) Distance-$r$ Dominating Set problem on $\mathcal{C}$ for any $\alpha>1$ (assuming ${FPT}\neq{W}[1]$).

How Much Does a Treedepth Modulator Help to Obtain Polynomial Kernels Beyond Sparse Graphs?

In the last years, kernelization with structural parameters has been an active area of research within the field of parameterized complexity. As a relevant example, Gajarský et al. (J Comput Syst Sci 84:219–242, 2017) proved that every graph problem satisfying a property called finite integer index admits a linear kernel on graphs of bounded expansion and an almost linear kernel on nowhere dense graphs, parameterized by the size of a c-treedepth modulator, which is a vertex set whose removal results in a graph of treedepth at most c, where $$c \ge 1$$ is a fixed integer. The authors left as further research to investigate this parameter on general graphs, and in particular to find problems that, while admitting polynomial kernels on sparse graphs, behave differently on general graphs. In this article we answer this question by finding two very natural such problems: we prove that Vertex Cover admits a polynomial kernel on general graphs for any integer $$c \ge 1$$ , and that Dominating Set does not for any integer $$c \ge 2$$ even on degenerate graphs, unless $$\text {NP} \subseteq \text {coNP}/\text {poly}$$ . For the positive result, we build on the techniques of Jansen and Bodlaender (Proceedings of the 28th symposium on theoretical aspects of computer science (STACS), volume 9 of LIPIcs, pp 177–188, 2011), and for the negative result we use a polynomial parameter transformation for $$c\ge 3$$ and an or-cross-composition for $$c = 2$$ . As existing results imply that Dominating Set admits a polynomial kernel on degenerate graphs for $$c = 1$$ , our result provides a dichotomy about the existence of polynomial kernels for Dominating Set on degenerate graphs with this parameter.

G-Convex Weight Sequences

In this paper, we introduce the notion of g-convex weight sequence (gcws) for connected graphs based on the concept of g-convexity and g-weight. g-weight is a natural generalization of the notion of branch weight for trees. We investigate the various questions of realization of an integer sequence as a g-convex weight sequence for trees and some special classes of graphs such as complete graphs, windmill and degenerate windmill graphs and wheels.

Hyperelliptic graphs and the period mapping on outer space

The period mapping assigns to each rank n, marked metric graph Gamma a positive definite quadratic form on H_1(Gamma). This defines maps Phi* and Phi on Culler--Vogtmann's outer space CV_n, and its Torelli space quotient T_n, respectively. The map Phi is a free group analog of the classical period mapping that sends a marked Riemann surface to its Jacobian. In this paper, we analyze the fibers of Phi in T_n, showing that they are aspherical, pi_1-injective subspaces. Metric graphs admitting a 'hyperelliptic involution' play an important role in the structure of Phi, leading us to define the hyperelliptic Torelli group, ST(n) < Out(F_n). We obtain generators for ST(n), and apply them to show that the connected components of the locus of 'hyperelliptic' graphs in T_n become simply-connected when certain degenerate graphs at infinity are added.

Some bounds on the generalised total chromatic number of degenerate graphs

The total generalised colourings considered in this paper are colourings of the vertices and of the edges of graphs satisfying the following conditions:•each set of vertices of the graph which receive the same colour induces an m-degenerate graph,•each set of edges of the graph which receive the same colour induces an n-degenerate graph, and•incident elements receive different colours. Bounds for the least number of colours with which this can be done for all k-degenerate graphs are obtained.

Bayesian selection of graphical regulatory models

We define a new class of coloured graphical models, called regulatory graphs. These graphs have their own distinctive formal semantics and can directly represent typical qualitative hypotheses about regulatory processes like those described by various biological mechanisms. They admit an embellishment into classes of probabilistic statistical models and so standard Bayesian methods of model selection can be used to choose promising candidate explanations of regulation. Regulation is modelled by the existence of a deterministic relationship between the longitudinal series of observations labelled by the receiving vertex and the donating one. This class contains longitudinal cluster models as a degenerate graph. Edge colours directly distinguish important features of the mechanism like inhibition and excitation and graphs are often cyclic. With appropriate distributional assumptions, because the regulatory relationships map onto each other through a group structure, it is possible to define a conditional conjugate analysis. This means that even when the model space is huge it is nevertheless feasible, using a Bayesian MAP search, to a discover regulatory network with a high Bayes Factor score. We also show that, like the class of Bayesian Networks, regulatory graphs also admit a formal but distinctive causal algebra. The topology of the graph then represents collections of hypotheses about the predicted effect of controlling the process by tearing out message passers or forcing them to transmit certain signals. We illustrate our methods on a microarray experiment measuring the expression of thousands of genes as a longitudinal series where the scientific interest lies in the circadian regulation of these plants.

From Edge-Coloring to Strong Edge-Coloring

In this paper we study a generalization of both proper edge-coloring and strong edge-coloring: $k$-intersection edge-coloring, introduced by Muthu, Narayanan and Subramanian. In this coloring, the set $S(v)$ of colors used by edges incident to a vertex $v$ does not intersect $S(u)$ on more than $k$ colors when $u$ and $v$ are adjacent. We provide some sharp upper and lower bounds for $\chi'_{k\text{-int}}$ for several classes of graphs. For $l$-degenerate graphs we prove that $\chi'_{k\text{-int}}(G)\leq (l+1)\Delta -l(k-1)-1$. We improve this bound for subcubic graphs by showing that $\chi'_{2\text{-int}}(G)\leq 6$. We show that calculating $\chi'_{k\text{-int}}(K_n)$ for arbitrary values of $k$ and $n$ is related to some problems in combinatorial set theory and we provide bounds that are tight for infinitely many values of $n$. Furthermore, for complete bipartite graphs we prove that $\chi'_{k\text{-int}}(K_{n,m}) = \left\lceil \frac{mn}{k}\right\rceil$. Finally, we show that computing $\chi'_{k\text{-int}}(G)$ is NP-complete for every $k\geq 1$.An addendum was added to this paper on Jul 4, 2015.

On vertex rankings of graphs and its relatives

A vertex ranking of a graph is an assignment of ranks (or colors) to the vertices of the graph, in such a way that any simple path connecting two vertices of equal rank, must contain a vertex of a higher rank. In this paper we study a relaxation of this notion, in which the requirement above should only hold for paths of some bounded length l for some fixed l. For instance, already the case l=2 exhibits quite a different behavior than proper coloring. We prove upper and lower bounds on the minimum number of ranks required for several graph families, such as trees, planar graphs, graphs excluding a fixed minor and degenerate graphs.

Parameterized complexity of three edge contraction problems with degree constraints

For any graph class $$\mathcal{H}$$ H , the $$\mathcal{H}$$ H -Contraction problem takes as input a graph $$G$$ G and an integer $$k$$ k , and asks whether there exists a graph $$H\in \mathcal{H}$$ H ? H such that $$G$$ G can be modified into $$H$$ H using at most $$k$$ k edge contractions. We study the parameterized complexity of $$\mathcal{H}$$ H -Contraction for three different classes $$\mathcal{H}$$ H : the class $$\mathcal{H}_{\le d}$$ H ≤ d of graphs with maximum degree at most $$d$$ d , the class $$\mathcal{H}_{=d}$$ H = d of $$d$$ d -regular graphs, and the class of $$d$$ d -degenerate graphs. We completely classify the parameterized complexity of all three problems with respect to the parameters $$k$$ k , $$d$$ d , and $$d+k$$ d + k . Moreover, we show that $$\mathcal{H}$$ H -Contraction admits an $$O(k)$$ O ( k ) vertex kernel on connected graphs when $$\mathcal{H}\in \{\mathcal{H}_{\le 2},\mathcal{H}_{=2}\}$$ H ? { H ≤ 2 , H = 2 } , while the problem is $$\mathsf{W}[2]$$ W [ 2 ] -hard when $$\mathcal{H}$$ H is the class of $$2$$ 2 -degenerate graphs and hence is expected not to admit a kernel at all. In particular, our results imply that $$\mathcal{H}$$ H -Contraction admits a linear vertex kernel when $$\mathcal{H}$$ H is the class of cycles.

Convex geometric [formula omitted]-quasiplanar representations of semi-bar [formula omitted]-visibility graphs

Semi-bar k-visibility graphs are graphs for which vertices can be drawn as horizontal segments with left endpoints on the y-axis (semi-bars) and edges can be drawn as vertical segments (sightlines) so that two semi-bars are visible to each other if and only if there is a sightline which intersects them and at most k other semi-bars. Cyclic semi-bar k-visibility graphs are graphs for which vertices can be drawn as segments with one endpoint at the origin (semi-bars) and edges can be drawn as arcs of circles centered at the origin (sightlines) so that two semi-bars are visible to each other if and only if there is a sightline which intersects them and at most k other semi-bars. We show that every semi-bar or cyclic semi-bar k-visibility graph can be represented in the plane with vertices drawn as points in convex position and edges drawn as segments so that there are no k+2 pairwise crossing edges. Furthermore, we prove that the class of graphs having cyclic semi-bar k-visibility representations with semi-bars of different lengths is the same as the class of (2k+2)-degenerate graphs that can be represented in the plane with vertices drawn as points in convex position and edges drawn as segments so that there are no k+2 pairwise crossing edges and so that adding any edge would result in k+2 pairwise crossing edges.