Class Of Subgraphs Research Articles

B0-VPG Representation of AT-free Outerplanar Graphs

A $k$-bend path is a non-self-intersecting polyline in the plane made of at most $k+1$ axis-parallel line segments. B$_{k}$-VPG is the class of graphs which can be represented as intersection graphs of $k$-bend paths in the same plane. In this paper, we show that all AT-free outerplanar graphs are B$_{0}$-VPG, i.e., intersection graphs of horizontal and vertical line segments in the plane. Our proofs are constructive and give a polynomial time B$_{0}$-VPG drawing algorithm for the class. In fact, we show the existence of a B$_{0}$-VPG representation for a superclass of AT-free graphs namely linear outerplanar graphs which we define as the class of subgraphs of biconnected outerpaths. Outerpaths are outerplanar graphs which admit a planar drawing whose weak dual is a path. Following a long line of improvements, Gonçalves, Isenmann, and Pennarun [SODA 2018] showed that all planar graphs are B$_{1}$-VPG. Since there are planar graphs which are not B$_{0}$-VPG, characterizing B$_{0}$-VPG graphs among planar graphs becomes interesting. Chaplick et al. [WG 2012] had shown that it is NP-complete to recognize B$_{k}$-VPG graphs within B$_{k+1}$-VPG. Hence recognizing B$_{0}$-VPG graphs within B$_{1}$-VPG is NP-complete in general, but the question is open when restricted to planar graphs. There are outerplanar graphs and AT-free planar graphs which are not B$_{0}$-VPG. This piqued our interest in AT-free outerplanar graphs.

A note on classes of subgraphs of locally finite graphs

We investigate the question how ‘small’ a graph can be, if it contains all members of a given class of locally finite graphs as subgraphs or induced subgraphs. More precisely, we give necessary and sufficient conditions for the existence of a connected, locally finite graph H containing all elements of a graph class G. These conditions imply that such a graph H exists for the class Gd consisting of all graphs with maximum degree <d which raises the question whether in this case H can be chosen to have bounded maximum degree. We show that this is not the case, thereby answering a question recently posed by Huynh et al.

Identifying the Exact Value of the Metric Dimension and Edge Dimension of Unicyclic Graphs

Given a simple connected graph G, the metric dimension dim(G) (and edge metric dimension edim(G)) is defined as the cardinality of a smallest vertex subset S⊆V(G) for which every two distinct vertices (and edges) in G have distinct distances to a vertex of S. It is an interesting topic to discuss the relation between these two dimensions for some class of graphs. This paper settles two open problems on this topic for unicyclic graphs. We recently learned that Sedlar and Škrekovski settled these problems, but our work presents the results in a completely different way. By introducing four classes of subgraphs, we characterize the structure of a unicyclic graph G such that dim(G) and edim(G) are equal to the cardinality of any minimum branch-resolving set for unicyclic graphs. This generates an approach to determine the exact value of the metric dimension (and edge metric dimension) for a unicyclic graph.

The poset of unlabeled induced subgraphs of a finite graph

The set of all unlabeled induced subgraphs of a finite graph G can be made into a poset by defining $$H_1 \le H_2$$ iff $$H_1$$ is a subgraph of $$H_2$$. This paper discusses some of the connections between the graph properties of G and the order theoretic properties of this poset. This paper then restricts this class of subgraphs to only connected unlabeled induced subgraphs and looks at some of the order theoretic properties of that poset.

The Graph Theory General Position Problem on Some Interconnection Networks

Given a graph $G$, the (graph theory) general position problem is to find the maximum number of vertices such that no three vertices lie on a common geodesic. This graph invariant is called the general position number (gp-number for short) of $G$ and denoted by ${\rm gp}(G)$. In this paper, the gp-number is determined for a large class of subgraphs of the infinite grid graph and for the infinite diagonal grid. To derive these results, we introduce monotone-geodesic labeling and prove a Monotone Geodesic Lemma that is in turn developed using the Erdos-Szekeres theorem on monotone sequences. The gp-number of the 3-dim infinite grid is bounded. Using isometric path covers, the gp-number is also determined for Benes networks.

Linear-time superbubble identification algorithm for genome assembly

DNA sequencing is the process of determining the exact order of the nucleotide bases of an individual's genome in order to catalogue sequence variation and understand its biological implications. Whole-genome sequencing techniques produce masses of data in the form of short sequences known as reads. Assembling these reads into a whole genome constitutes a major algorithmic challenge. Most assembly algorithms utilise de Bruijn graphs constructed from reads for this purpose. A critical step of these algorithms is to detect typical motif structures in the graph caused by sequencing errors and genome repeats, and filter them out; one such complex subgraph class is a so-called superbubble. In this paper, we propose an O(n+m)-time algorithm to detect all superbubbles in a directed acyclic graph with n vertices and m (directed) edges, improving the best-known O(mlog⁡m)-time algorithm by Sung et al.

Some Results on the Eigenvalues of Distance-Regular Graphs

In 1986, Terwilliger showed that there is a strong relation between the eigenvalues of a distance-regular graph and the eigenvalues of a local graph. In particular, he showed that the eigenvalues of a local graph are bounded in terms of the eigenvalues of a distance-regular graph, and he also showed that if an eigenvalue $$\theta $$? of the distance-regular graph has multiplicity m less than its valency k, then $$-1- \frac{b_1}{\theta +1}$$-1-b1?+1 is an eigenvalue for any local graph with multiplicity at least $$k-m$$k-m. In this paper, we are going to generalize the results of Terwilliger to a broader class of subgraphs. Instead of local graphs, we consider induced subgraphs of distance-regular graphs (and more generally t-walk-regular graphs with t a positive integer) which satisfies certain regularity conditions. Then we apply the results to obtain bounds on eigenvalues of t-walk-regular graphs if the girth equals 3, 4, 5, 6 or 8. In particular, we will show that the second largest eigenvalue of a distance-regular graph with girth 6 and valency k is at most $$k-1$$k-1, and we will show that the only such graphs having $$k-1$$k-1 as its second largest eigenvalue are the doubled Odd graphs.

An O(m log m)-Time Algorithm for Detecting Superbubbles.

In genome assembly graphs, motifs such as tips, bubbles, and cross links are studied in order to find sequencing errors and to understand the nature of the genome. Superbubble, a complex generalization of bubbles, was recently proposed as an important subgraph class for analyzing assembly graphs. At present, a quadratic time algorithm is known. This paper gives an O(m log m)-time algorithm to solve this problem for a graph with m edges.

Limiting geodesics for first-passage percolation on subsets of $\mathbb{Z}^{2}$

It is an open problem to show that in two-dimensional first-passage percolation, the sequence of finite geodesics from any point to $(n,0)$ has a limit in $n$. In this paper, we consider this question for first-passage percolation on a wide class of subgraphs of $\mathbb {Z}^2$: those whose vertex set is infinite and connected with an infinite connected complement. This includes, for instance, slit planes, half-planes and sectors. Writing $x_n$ for the sequence of boundary vertices, we show that the sequence of geodesics from any point to $x_n$ has an almost sure limit assuming only existence of finite geodesics. For all passage-time configurations, we show existence of a limiting Busemann function. Specializing to the case of the half-plane, we prove that the limiting geodesic graph has one topological end; that is, all its infinite geodesics coalesce, and there are no backward infinite paths. To do this, we prove in the Appendix existence of geodesics for all product measures in our domains and remove the moment assumption of the Wehr-Woo theorem on absence of bigeodesics in the half-plane.

An algorithm for score aggregation over causal biological networks based on random walk sampling.

BackgroundWe recently published in BMC Systems Biology an approach for calculating the perturbation amplitudes of causal network models by integrating gene differential expression data. This approach relies on the process of score aggregation, which combines the perturbations at the level of the individual network nodes into a global measure that quantifies the perturbation of the network as a whole. Such "bottom-up" aggregation relates the changes in molecular entities measured by omics technologies to systems-level phenotypes. However, the aggregation method we used is limited to a specific class of causal network models called "causally consistent", which is equivalent to the notion of balance of a signed graph used in graph theory. As a consequence of this limitation, our aggregation method cannot be used in the many relevant cases involving "causally inconsistent" network models such as those containing negative feedbacks.FindingsIn this note, we propose an algorithm called "sampling of spanning trees" (SST) that extends our published aggregation method to causally inconsistent network models by replacing the signed relationships between the network nodes by an appropriate continuous measure. The SST algorithm is based on spanning trees, which are a particular class of subgraphs used in graph theory, and on a sampling procedure leveraging the properties of specific random walks on the graph. This algorithm is applied to several cases of biological interest.ConclusionsThe SST algorithm provides a practical means of aggregating nodal values over causally inconsistent network models based on solid mathematical foundations. We showed its utility in systems biology, where the nodal values can be perturbation amplitudes of protein activities or gene differential expressions, while the networks can be models of cellular signaling or expression regulation. Since the SST algorithm is based on general graph-theoretical considerations, it is scalable to arbitrary graph sizes and can potentially be used for performing quantitative analyses in any context involving signed graphs.Electronic supplementary materialThe online version of this article (doi:10.1186/1756-0500-7-516) contains supplementary material, which is available to authorized users.

An Extremal Theorem in the Hypercube

The hypercube $Q_n$ is the graph whose vertex set is $\{0,1\}^n$ and where two vertices are adjacent if they differ in exactly one coordinate. For any subgraph $H$ of the cube, let $\textrm{ex}(Q_n, H)$ be the maximum number of edges in a subgraph of $Q_n$ which does not contain a copy of $H$. We find a wide class of subgraphs $H$, including all previously known examples, for which $\textrm{ex}(Q_n, H) = o(e(Q_n))$. In particular, our method gives a unified approach to proving that $\textrm{ex}(Q_n, C_{2t}) = o(e(Q_n))$ for all $t \geq 4$ other than $5$.

Maxmaxflow and Counting Subgraphs

We introduce a new graph invariant $\Lambda(G)$ that we call maxmaxflow, and put it in the context of some other well-known graph invariants, notably maximum degree and its relatives. We prove the equivalence of two "dual" definitions of maxmaxflow: one in terms of flows, the other in terms of cocycle bases. We then show how to bound the total number (or more generally, total weight) of various classes of subgraphs of $G$ in terms of either maximum degree or maxmaxflow. Our results are motivated by a conjecture that the modulus of the roots of the chromatic polynomial of $G$ can be bounded above by a function of $\Lambda(G)$.

Local Computations on Triangular Graphs

The paper deals with the class of finite triangular graphs. It turns out that this class enjoys regular properties similar to those of trees and complete graphs. The main objective of the paper is to lift algorithms for some typical local computations, known for other classes of graphs, to the class of triangular graphs. Local algorithms on graphs, according to [8, 9], are defined as local rules for relabeling graph nodes. Rules are local, if they are defined only for a class of subgraphs of processed graph (as neighborhoods of nodes or edges) and neither their results nor their applicability do not depend upon the knowledge of the whole graph labeling. While designing local algorithm for triangular graphs one needs to use some intrinsic properties of such graphs; it puts some additional light on their inherent structure. To illustrate essential features of local computations on triangular graphs, local algorithms for three typical issues of local computations are discussed: leader election, spanning tree construction, and nodes ordering. Correctness of these algorithms, as deadlock freeness, proper termination, and impartiality, are proved.

Pairs of forbidden class of subgraphs concerning K1,3and P6to have a cycle containing specified vertices

Pairs of forbidden class of subgraphs concerning K_1,3and P₆to have a cycle containing specified vertices

The topological relationship between the large-scale attributes and local interaction patterns of complex networks

Recent evidence indicates that the abundance of recurring elementary interaction patterns in complex networks, often called subgraphs or motifs, carry significant information about their function and overall organization. Yet, the underlying reasons for the variable quantity of different subgraph types, their propensity to form clusters, and their relationship with the networks' global organization remain poorly understood. Here we show that a network's large-scale topological organization and its local subgraph structure mutually define and predict each other, as confirmed by direct measurements in five well studied cellular networks. We also demonstrate the inherent existence of two distinct classes of subgraphs, and show that, in contrast to the low-density type II subgraphs, the highly abundant type I subgraphs cannot exist in isolation but must naturally aggregate into subgraph clusters. The identified topological framework may have important implications for our understanding of the origin and function of subgraphs in all complex networks.

Moderate deviations for functional U-processes

The moderate deviations principle is shown for the partial sums processes built of U-empirical measures and of U-statistics. It is proved that in the non-degenerate case the conditions for the fixed time principles suffice for the moderate deviations principle to carry over to the corresponding partial sums processes. Given a uniformly bounded VC subgraph class of functions, we obtain corresponding moderate deviations for time dependent U-processes. We use decoupling techniques and apply an improved version of a Bernstein-type inequality for degenerate U-statistics. Moreover, we prove and use a Lévy-type maximal inequality for U-statistics.

Compound Poisson approximations of subgraph counts in random graphs

Poisson approximation, random graphs, Stein's method Poisson approximations for the counts of a given subgraph in large random graphs were accomplished using Stein's method by Barbour and others. Compound Poisson approximation results, on the other hand, have not appeared, at least partly because of the lack of a suitable coupling. We address that problem by introducing the concept of cluster determining pairs, leading to a useful coupling for a large class of subgraphs we call local. We find bounds on the compound Poisson approximation of counts of local subgraphs in large random graphs.

Combinatorics of RNA secondary structures

Secondary structures of polynucleotides can be viewed as a class of planar vertex-labeled graphs. We compute recursion formulae for enumerating a variety sub-classes of and classes of sub-graphs (structural elements) of secondary structure graphs. First order asymptotics are derived and their dependence on the composition of the underlying nucleic acid sequences is discussed.

Moderate and large deviations for U-processes

Sufficient conditions for the moderate and large deviation principle for U-processes are given. For the large deviation result the conditions are in terms of “blockwise” empirical process conditions. On the moderate scaling the case of U-processes indexed by a uniformly bounded VC subgraph class of functions is considered. The proofs are based on an isoperimetric inequality for empirical processes due to Talagrand, a truncation method based on an isoperimetric inequality by Ledoux, the existence of almost regular partitions of complete hypergraphs due to Baranyai and a Bernstein-type inequality for U-processes due to Arcones and Giné.

The low of the iterated logarithm for empirical processes under absolute regularity

The compact law of the iterated logarithm for empirical processes whose underlying sequence satisfies a β-mixing condition is considered. In particular, we show a compact law of the iterated logarithm for VC subgraph classes of functions, for classes of functions which satisfy the bracketing condition in Doukhanet al. (6) and for some classes of smooth functions.