Arboricity Of Graph Research Articles

Near-optimal distributed dominating set in bounded arboricity graphs

AbstractWe describe a simple deterministic $$O( \varepsilon ^{-1} \log \Delta )$$ O ( ε - 1 log Δ ) round distributed algorithm for $$(2\alpha +1)(1 + \varepsilon )$$ ( 2 α + 1 ) ( 1 + ε ) approximation of minimum weighted dominating set on graphs with arboricity at most $$\alpha $$ α . Here $$\Delta $$ Δ denotes the maximum degree. We also show a lower bound proving that this round complexity is nearly optimal even for the unweighted case, via a reduction from the celebrated KMW lower bound on distributed vertex cover approximation (Kuhn et al. in JACM 63:116, 2016). Our algorithm improves on all the previous results (that work only for unweighted graphs) including a randomized $$O(\alpha ^2)$$ O ( α 2 ) approximation in $$O(\log n)$$ O ( log n ) rounds (Lenzen et al. in International symposium on distributed computing, Springer, 2010), a deterministic $$O(\alpha \log \Delta )$$ O ( α log Δ ) approximation in $$O(\log \Delta )$$ O ( log Δ ) rounds (Lenzen et al. in international symposium on distributed computing, Springer, 2010), a deterministic $$O(\alpha )$$ O ( α ) approximation in $$O(\log ^2 \Delta )$$ O ( log 2 Δ ) rounds (implicit in Bansal et al. in Inform Process Lett 122:21–24, 2017; Proceeding 17th symposium on discrete algorithms (SODA), 2006), and a randomized $$O(\alpha )$$ O ( α ) approximation in $$O(\alpha \log n)$$ O ( α log n ) rounds (Morgan et al. in 35th International symposiumon distributed computing, 2021). We also provide a randomized $$O(\alpha \log \Delta )$$ O ( α log Δ ) round distributed algorithm that sharpens the approximation factor to $$\alpha (1+o(1))$$ α ( 1 + o ( 1 ) ) . If each node is restricted to do polynomial-time computations, our approximation factor is tight in the first order as it is NP-hard to achieve $$\alpha - 1 - \varepsilon $$ α - 1 - ε approximation (Bansal et al. in Inform Process Lett 122:21-24, 2017).

An estimator for matching size in low arboricity graphs with two applications

In this paper, we present a new degree-based estimator for the size of maximum matching in bounded arboricity graphs. When the arboricity of the graph is bounded by \(\alpha \), the estimator gives a \(\alpha +2\) factor approximation of the matching size. For planar graphs, we show the estimator does better and returns a 3.5 approximation of the matching size. Using this estimator, we get new results for approximating the matching size of planar graphs in the streaming and distributed models of computation. In particular, in the vertex-arrival streams, we get a randomized \(O\left( \frac{\sqrt{n}}{\varepsilon ^2}\log n\right) \) space algorithm for approximating the matching size of a planar graph on n vertices within \((3.5+\varepsilon )\) factor. Similarly, we get a simultaneous protocol in the vertex-partition model for approximating the matching size within \((3.5+\varepsilon )\) factor using \(O\left( \frac{n^{2/3}}{\varepsilon ^2}\log n\right) \) communication from each player. In comparison with previous works, the estimator in this paper does not need to know the arboricity of the input graph and improves the approximation factor in the case of planar graphs.

Edge-disjoint spanning trees and forests of graphs

The spanning tree packing number of a graph G is the maximum number of edge-disjoint spanning trees in G, and the arboricity of G is the minimum number of edge-disjoint forests needed to partition the edge set of G. In this paper, we give bounds on the spanning tree packing number and the arboricity of graphs in terms of effective resistances. As applications, we show that equiarboreal graphs are uniformly dense, and determine the maximum number of edge-disjoint spanning c-forests of equiarboreal graphs, including edge-transitive graphs, 1-walk-regular graphs, distance-regular graphs and so on. The arboricity of a regular graph can be derived from our effective resistance bounds directly.

Vertex arboricity of graphs embedded in a surface of non-negative Euler characteristic

The vertex arboricity [Formula: see text] of a graph [Formula: see text] is the minimum number of colors the vertices of the graph [Formula: see text] can be colored so that every color class induces an acyclic subgraph of [Formula: see text]. There are many results on the vertex arboricity of planar graphs. In this paper, we replace planar graphs with graphs which can be embedded in a surface [Formula: see text] of Euler characteristic [Formula: see text]. We prove that for the graph [Formula: see text] which can be embedded in a surface [Formula: see text] of Euler characteristic [Formula: see text] if no [Formula: see text]-cycle intersects a [Formula: see text]-cycle, or no [Formula: see text]-cycle intersects a [Formula: see text]-cycle, then [Formula: see text] in addition to the [Formula: see text]-regular quadrilateral mesh.

Improved Dynamic Graph Coloring

This article studies the fundamental problem of graph coloring in fully dynamic graphs. Since the problem of computing an optimal coloring, or even approximating it to within n1-εfor any ε > 0, is NP-hard in static graphs, there is no hope to achieve any meaningful computational results for general graphs in the dynamic setting. It is therefore only natural to consider the combinatorial aspects of dynamic coloring or alternatively, study restricted families of graphs.Toward understanding the combinatorial aspects of this problem, one may assume a black-box access to a static algorithm forC-coloring any subgraph of the dynamic graph, and investigate the trade-off between the number of colors and the number of recolorings per update step. Optimizing the number of recolorings, sometimes referred to as the recourse bound, is important for various practical applications. In WADS ’17, Barba et al. devised two complementary algorithms: for any β > 0, the first (respectively, second) maintains anO(Cβn1/β)(respectively,O(Cβ)-coloring while recoloring O(β) (respectively, O(βn1/β)) vertices per update. Barba et al. also showed that the second trade-off appears to exhibit the right behavior, at least for β = O(1): any algorithm that maintains aC-coloring of ann-vertex dynamic forest must recolor Ω (n2C(C-1)) vertices per update, for any constant C ≥ 2. Our contribution is twofold:• We devise a new algorithm for general graphs that improves significantly upon the first trade-off in a wide range of parameters: for any β > 0, we get a Ô (Cβlog2n)-coloring with O(β) recolorings per update, where the Ô notation suppressespolyloglog(n)factors. In particular, for β = O(1), we get constant recolorings withpolylog(n)colors; not only is this an exponential improvement over the previous bound but also it unveils a rather surprising phenomenon: the trade-off between the number of colors and recolorings is highly non-symmetric.• For uniformly sparse graphs, we use low out-degree orientations to strengthen the preceding result by bounding the update time of the algorithm rather than the number of recolorings. Then, we further improve this result by introducing a new data structure that refines bounded out-degree edge orientations and is of independent interest. From this data structure, we get a deterministic algorithm for graphs of arboricity ɑ that maintains an O(ɑ log2n)-coloring in amortized O(1) time.

Fully Dynamic MIS in Uniformly Sparse Graphs

We consider the problem of maintaining a maximal independent set in a dynamic graph subject to edge insertions and deletions. Recently, Assadi et al. (at STOC’18) showed that a maximal independent set can be maintained in sublinear (in the dynamically changing number of edges) amortized update time. In this article, we significantly improve the update time for uniformly sparse graphs . Specifically, for graphs with arboricity α, the amortized update time of our algorithm is O (α 2 ⋅ log 2 n ), where n is the number of vertices. For low arboricity graphs, which include, for example, minor-free graphs and some classes of “real-world” graphs, our update time is polylogarithmic. Our update time improves the result of Assadi et al. for all graphs with arboricity bounded by m 3/8−ϵ , for any constant ϵ > 0. This covers much of the range of possible values for arboricity, as the arboricity of a general graph cannot exceed m 1/2 .

Simple Deterministic Algorithms for Fully Dynamic Maximal Matching

A maximal matching can be maintained in fully dynamic (supporting both addition and deletion of edges) n -vertex graphs using a trivial deterministic algorithm with a worst-case update time of O ( n ). No deterministic algorithm that outperforms the naïve O ( n ) one was reported up to this date. The only progress in this direction is due to Ivković and Lloyd, who in 1993 devised a deterministic algorithm with an amortized update time of O (( n + m ) √2/2 ), where m is the number of edges. In this article, we show the first deterministic fully dynamic algorithm that outperforms the trivial one. Specifically, we provide a deterministic worst-case update time of O (√ m ). Moreover, our algorithm maintains a matching, which in fact is a 3/2-approximate maximum cardinality matching (MCM). We remark that no fully dynamic algorithm for maintaining (2 − ϵ)-approximate MCM improving upon the naïve O ( n ) was known prior to this work, even allowing amortized time bounds and randomization. For low arboricity graphs (e.g., planar graphs and graphs excluding fixed minors), we devise another simple deterministic algorithm with sublogarithmic update time. Specifically, it maintains a fully dynamic maximal matching with amortized update time of O (log n /log log n ). This result addresses an open question of Onak and Rubinfeld [2010]. We also show a deterministic algorithm with optimal space usage, which for arbitrary graphs maintains a maximal matching in amortized O (√ m ) time and uses only O ( n + m ) space.

On the equitable vertex arboricity of graphs

An equitable -tree-colouring of a graph G is a t-colouring of vertices of G such that the sizes of any two colour classes differ by at most one and the subgraph induced by each colour class is a forest of maximum degree at most k. The strong equitable vertex k-arboricity, denoted by , is the smallest t such that G has an equitable-tree-colouring for every . In this paper, we give upper bounds for when G is a balanced complete bipartite graph and . For some special cases, we determine the exact values. We also prove that: (1) for every planar graph without 4-cycles, 5-cycles and 6-cycles; (2) for every planar graph with neither 3-cycles nor adjacent 4-cycles.

Degree sequence realizations with given packing and covering of spanning trees

Designing networks in which every processor has a given number of connections often leads to graphic degree sequence realization models. A nonincreasing sequence d=(d1,d2,…,dn) is graphic if there is a simple graph G with degree sequence d. The spanning tree packing number of graph G, denoted by τ(G), is the maximum number of edge-disjoint spanning trees in G. The arboricity of graph G, denoted by a(G), is the minimum number of spanning trees whose union covers E(G). In this paper, it is proved that, given a graphic sequence d=d1≥d2≥⋯≥dn and integers k2≥k1>0, there exists a simple graph G with degree sequence d satisfying k1≤τ(G)≤a(G)≤k2 if and only if dn≥k1 and 2k1(n−1)≤∑i=1ndi≤2k2(n−|I|−1)+2∑i∈Idi, where I={i:di<k2}. As corollaries, for any integer k>0, we obtain a characterization of graphic sequences with at least one realization G satisfying a(G)≤k, and a characterization of graphic sequences with at least one realization G satisfying τ(G)=a(G)=k.

A Weaker Version of a Conjecture on List Vertex Arboricity of Graphs

The vertex arboricity $$\rho (G)$$?(G) of a graph $$G$$G is the minimum number of colors to color $$G$$G such that each color class induces a forest. The list vertex arboricity $$\rho _l(G)$$?l(G) is the list-coloring version of this concept. Zhen and Wu conjectured that $$\rho _l(G)=\rho (G)$$?l(G)=?(G) whenever $$|V(G)|\le 3\rho (G)$$|V(G)|≤3?(G). In this paper, we prove the weaker version of the conjecture obtained by replacing $$3\rho (G)$$3?(G) with $$\frac{5}{2}\rho (G)+\frac{1}{2}$$52?(G)+12.

Arboricity, [formula omitted]-index, and dynamic algorithms

We propose a new data structure for manipulating graphs, called h-graph, which is particularly suited for designing dynamic algorithms. The structure itself is simple, consisting basically of a triple of elements, for each vertex of the graph. The overall size of all triples is O(n+m), for a graph with n vertices and m edges. We describe algorithms for performing the basic operations related to dynamic applications, as insertions and deletions of vertices or edges, and adjacency queries. The data structure employs a technique first described by Chiba and Nishizeki [Chiba, Nishizeki, Arboricity and subgraph listing algorithms, SIAM J. Comput. 14 (1) (1985) 210–223], and relies on the arboricity of graphs. Using the proposed data structure, we describe several dynamic algorithms for solving problems as listing the cliques of a given size, recognizing diamond-free graphs, and finding simple, simplicial and dominated vertices. These algorithms are the first of their kind to be proposed in the literature. In fact, the dynamic algorithms for the above problems lead directly to new static algorithms, and using the data structure we also design new static algorithms for the problems of counting subgraphs of size 4, recognizing cop-win graphs and recognizing strongly chordal graphs. The complexities of all of the proposed static algorithms improve over the complexities of the so far existing algorithms, for graphs of low arboricity. In addition, for the problems of counting subgraphs of size 4 and recognizing diamond-free graphs, the improvement is general.

On graph thickness, geometric thickness, and separator theorems

We investigate the relationship between geometric thickness, thickness, outerthickness, and arboricity of graphs. In particular, we prove that all graphs with arboricity two or outerthickness two have geometric thickness O ( log n ) . The technique used can be extended to other classes of graphs so long as a separator theorem exists. For example, we can apply it to show the known bound that thickness two graphs have geometric thickness O ( n ) , yielding a simple construction in the process.

The Linear Arboricity of Graphs on Surfaces of Negative Euler Characteristic

The linear arboricity of a graph $G$ is the minimum number of linear forests which partition the edges of $G$. In the present, it is proved that if a graph $G$ can be embedded in a surface of Euler characteristic $\varepsilon<0$ and $\Delta(G)\geq\sqrt{46-54\varepsilon}+19$, then its linear arboricity is $\lceil\frac{\Delta(G)}{2}\rceil$. Some related results on the girth and maximum average degree are also obtained.

Adjacency queries in dynamic sparse graphs

We deal with the problem of maintaining a dynamic graph so that queries of the form “is there an edge between u and v?” are processed fast. We consider graphs of bounded arboricity, i.e., graphs with no dense subgraphs, like, for example, planar graphs. Brodal and Fagerberg [G.S. Brodal, R. Fagerberg, Dynamic representations of sparse graphs, in: Proc. 6th Internat. Workshop on Algorithms and Data Structures (WADS'99), in: Lecture Notes in Comput. Sci., vol. 1663, Springer, Berlin, 1999, pp. 342–351] described a very simple linear-size data structure which processes queries in constant worst-case time and performs insertions and deletions in O ( 1 ) and O ( log n ) amortized time, respectively. We show a complementary result that their data structure can be used to get O ( log n ) worst-case time for query, O ( 1 ) amortized time for insertions and O ( 1 ) worst-case time for deletions. Moreover, our analysis shows that by combining the data structure of Brodal and Fagerberg with efficient dictionaries one gets O ( log log log n ) worst-case time bound for queries and deletions and O ( log log log n ) amortized time for insertions, with size of the data structure still linear. This last result holds even for graphs of arboricity bounded by O ( log k n ) , for some constant k.

Competitive Call Control in Mobile Networks

In this paper we deal with competitive local on-line algorithms for non-preemptive channel allocation in mobile networks. The signal interferences in a network are modeled using an interference graph G . We prove that the greedy on-line algorithm is Δ -competitive, where Δ is the maximum degree of G . We employ the ``classify and randomly select" paradigm [5], [17], and give a 5 -competitive randomized algorithm for the case of planar interference graphs, a 2 -competitive randomized algorithm for trees, and a (2c) -competitive randomized algorithm for graphs of arboricity c . We also show that the problem of call control in mobile networks with multiple available frequencies reduces to the problem of call control in mobile networks with a single frequency. Using this reduction, we present on-line algorithms for general networks with a single frequency. We give a local on-line algorithm which is (α (δ +1 + α )/(1/2+α )2 )-competitive, where α is the independence number of G , and δ is the average degree of G . The above results hold in the case when the duration of each request is infinite, and the benefit the algorithm gains by accepting each request is equal to one. They are extended to handle requests of arbitrary durations, and arbitrary benefits.

A note on the arboricity of graphs

The well known theorem of Nash-Williams determines the graphs that are union ofk edge disjoint forests. The main result presented in this note is that any graph which is the union ofk edge disjoint forests is in fact a union ofk such forests in which if a vertex has degree at least 3 in one of the forests then its degree is positive in all the other forests. We also discuss consequences of this result with respect to the arboricity of regular graphs.

More results on Ramsey—Turán type problems

The paper deals with common generalizations of classical results of Ramsey and Turán. The following is one of the main results. Assumek≧2, ε>0,G n is a sequence of graphs ofn-vertices and at least 1/2((3k−5) / (3k−2)+ε)n 2 edges, and the size of the largest independent set inG n iso(n). LetH be any graph of arboricity at mostk. Then there exists ann 0 such that allG n withn>n 0 contain a copy ofH. This result is best possible in caseH=K 2k .