Graph Parameters Research Articles

Serial and parallel kernelization of Multiple Hitting Set parameterized by the Dilworth number, implemented on the GPU

The NP-hard Multiple Hitting Set problem is the problem of finding a minimum-cardinality set intersecting each of the sets in a given input collection a given number of times. Generalizing a well-known data reduction algorithm due to Weihe, we show a problem kernel for Multiple Hitting Set parameterized by the Dilworth number, a graph parameter introduced by Foldes and Hammer in 1978 yet seemingly so far unexplored in the context of parameterized complexity theory. Using matrix multiplication, we speed up the algorithm to quadratic sequential time and logarithmic parallel time. We experimentally evaluate our algorithms. By implementing our algorithm on GPUs, we show the feasibility of realizing kernelization algorithms on SIMD (Single Instruction, Multiple Data) architectures.

On the Lanzhou index of graphs

Let $G$ be a simple graph with vertex set $V(G)$ and edge set $E(G)$. The Lanzhou index of a graph $G$ is defined as ${\rm Lz}(G)=\sum_{u\in V(G)}\,d_{\overline{G}}(v)\,d_G(v)^2$, where $d_G(v)$ denotes the degree of the vertex $v$ in $G$. In this paper, we determine extremal values of the Lanzhou index in terms of some graph parameters, as well as Nordhaus--Gaddum--type results. We also find relations between Lanzhou Index and other topological indices.

Ex-utero third trimester developmental changes in functional brain network organization in infants born very and extremely preterm.

The latter half of gestation is a period of rapid brain development, including the formation of fundamental functional brain network architecture. Unlike in-utero fetuses, infants born very and extremely preterm undergo these critical maturational changes in the extrauterine environment, with growing evidence suggesting this may result in altered brain networks. To date, however, the development of functional brain architecture has been unexplored. From a prospective cohort of preterm infants, graph parameters were calculated for fMRI scans acquired prior to reaching term equivalent age. Eight graph properties were calculated, Clustering Coefficient (C), Characteristic Path Length (L), Modularity (Q), Local Efficiency (LE), Global Efficiency (GE), Normalized Clustering (λ), Normalized Path Length (γ), and Small-Worldness (σ). Properties were first compared to values generated from random and lattice networks and cost efficiency was evaluated. Subsequently, linear mixed effect models were used to assess relationship with postmenstrual age and infant sex. A total of 111 fMRI scans were acquired from 85 preterm infants born at a mean GA 28.93 ± 2.8. Infants displayed robust small world properties as well as both locally and globally efficient networks. Regression models found that GE increased while L, Q, λ, γ, and σ decreased with increasing postmenstrual age following multiple comparison correction (r2Adj range 0.143-0.401, p < 0048), with C and LE exhibited trending increases with age. This is the first direct investigation on the extra-uterine formation of functional brain architecture in preterm infants. Importantly, our results suggest that changes in functional architecture with increasing age exhibit a different trajectory relative to in utero fetus. Instead, they exhibit developmental changes more similar to the early postnatal period in term born infants.

Reconstruction from one labelled card and more

The deck D of a graph H is its multiset of one-vertex deleted subgraphs. Ulam's Reconstruction Conjecture claims that every graph having more than two vertices is reconstructible from D. We seek minimal graph parameters that need to accompany one card G to enable the unique reconstruction of H. A card needs to be feasible. We consider specific graph invariants associated with the 0–1 eigenspaces of H, which need to supplement a feasible card, for the unique reconstruction of H. Non–isomorphic graphs sharing a common card turn out to have linearly independent one-dimensional eigenspaces, whereas non–isomorphic graphs, sharing a common eigenspace, have no labelled card in common. Moreover, we determine certain families of graphs for which reconstruction is possible from any card.

Estrada index and subgraph centrality of hypergraphs via tensors

Uniform hypergraphs have a natural one-to-one correspondence to tensors. In this paper, we investigate the Estrada index and subgraph centrality of an m-uniform hypergraph H via the adjacency tensor. We establish some bounds for the Estrada index and give expressions of the subgraph centrality in terms of graph parameters of the multi-digraphs associated with H. When H is 2-uniform, the above Estrada index and subgraph centrality are the Estrada index and subgraph centrality of a graph.

Efficient parameterized algorithms for computing all-pairs shortest paths

Computing for all pairs of vertices the shortest paths in a graph is a fundamental and much-studied problem with many applications. Unfortunately, despite intense study, there are still no significantly faster algorithms for it than the O(n3) time algorithm due to Floyd and Warshall (1962). Somewhat faster algorithms exist for the vertex-weighted version if fast matrix multiplication may be used. Yuster (SODA 2009) gave an algorithm running in time O(n2.842), but no combinatorial, truly subcubic algorithm is known.Motivated by the recent framework of efficient parameterized algorithms (or “FPT in P”), we investigate the influence of the graph parameters clique-width (cw) and modular-width (mw) on the running times of algorithms for solving vertex-weighted all-pairs shortest paths. We obtain efficient (and combinatorial) parameterized algorithms of times O(cw2n2), resp. O(mw2n+n2). If fast matrix multiplication is allowed then the latter can be improved to O(mw1.842n+n2) using the algorithm of Yuster as a black box. The algorithm relative to modular-width is adaptive, meaning that the running time matches the best unparameterized algorithm for parameter value mw equal to n, and outperforms it already for mw∈O(n1−ɛ) for any ɛ>0.

DOMINATION PARAMETERS ON BIPARTITE GRAPHS OF THE COMMUTATIVE RING ZN

In 1977,E.J.Cockayne and S.T. Hedetniemi conducted a commendable and broad survey on the outcomes of the existing concepts of dominating set in graphs. It’s basic concept is the dominating set and the domination number. In this paper our main aim is to find out the dominating parameters of Bipartite graph obtained on the Commutative ring of type Zn. A subset D of V is said to be a dominating set of G if every vertex in V-D is adjacent to a vertex in D. A bipartite graph is a graph G whose vertex set is partitioned into two disjoint subsets X and Y such that each edge in G has one end in X and the other end in Y. We determine the domination parameters i.e., domination number, dominating set and Minimum domination number of Bipartite graphs of the commutative ring Zn.

How does the chromatic number of a random graph vary?

AbstractThe chromatic number of a graph is a fundamental parameter, whose study was originally motivated by applications ( is the minimum number of internally compatible groups the vertices can be divided into, if the edges represent incompatibility). As with other graph parameters, it is also studied from a purely theoretical point of view, and here a key question is: what is its typical value? More precisely, how does , the chromatic number of a graph chosen uniformly at random from all graphs on vertices, behave? This quantity is a random variable, so one can ask (i) for upper and lower bounds on its typical values, and (ii) for bounds on how much it varies: what is the width (for example, standard deviation) of its distribution? On (i) there has been considerable progress over the last 45 years; on (ii), which is our focus here, remarkably little. One would like both upper and lower bounds on the width of the distribution, and ideally a description of the (appropriately scaled) limiting distribution. There is a well‐known upper bound of Shamir and Spencer of order , improved slightly by Alon to , but no non‐trivial lower bound was known until 2019, when the first author proved that the width is at least for infinitely many , answering a longstanding question of Bollobás. In this paper we have two main aims: first, we shall prove a much stronger lower bound on the width. We shall show unconditionally that, for some values of , the width is at least , matching the upper bounds up to the error term. Moreover, conditional on a recently announced sharper explicit estimate for the chromatic number, we improve the lower bound to order , within a logarithmic factor of the upper bound. Second, we will describe a number of conjectures as to what the true behaviour of the variation in is, and why. The first form of this conjecture arises from recent work of Bollobás, Heckel, Morris, Panagiotou, Riordan and Smith. We will also give much more detailed conjectures, suggesting that the true width, for the worst case , matches our lower bound up to a constant factor. These conjectures also predict a Gaussian limiting distribution.

The brain functional connectivity alterations in traumatic patients with olfactory disorder after low-level laser therapy demonstrated by fMRI.

Low-level laser therapy (LLLT) has been clinically accepted to accelerate the nerve regeneration process after a nerve injury or transection. We aimed to investigate the neuronal basis and the influence of LLLT on brain functional networks in traumatic patients with olfactory dysfunction. Twenty-four Patients with traumatic anosmia/hyposmia were exposed to pleasant olfactory stimuli during a block-designed fMRI session. After a 10-week period, patients as control group and patients who had completed the sessions of LLLT were invited for follow-up testing using the same fMRI protocol. Two-sample t-tests were conducted to explore group differences in activation responding to odorants (p-FDR-corrected <0.05). Differences of functional connectivity were compared between the two groups and the topological features of the olfactory network were calculated. Correlation analysis was performed between graph parameters and TDI score. Compared to controls, laser-treated patients showed increased activation in the cingulate, rectus gyrus, and some parts of the frontal gyrus. Shorter pathlength (p = 0.047) and increased local efficiency (p = 0.043) within the olfactory network, as well as decreased inter-network connectivity within the whole brain were observed in patients after laser surgery. Moreover, higher clustering and local efficiency were related to higher TDI score, as manifested in increased sensitivity to identify odors. The results support that low-level laser induces neural reorganization process and make new connections in the olfactory structures. Furthermore, the connectivity parameters may serve as potential biomarkers for traumatic anosmia or hyposmia by revealing the underlying neural mechanisms of LLLT.

Some relations between energy and Seidel energy of a graph

AbstractThe energy E(G) of a graph G is the sum of the absolute values of eigenvalues of G and the Seidel energy ES(G) is the sum of the absolute values of eigenvalues of the Seidel matrix S of G. In this paper, some relations between the energy and Seidel energy of a graph in terms of different graph parameters are presented. Also, the inertia relations between the graph eigenvalue and Seidel eigenvalue of a graph are given. The results in this paper generalize some of the existing results.

Reducing Graph Parameters by Contractions and Deletions

We consider the following problem: for a given graph G and two integers k and d, can we apply a fixed graph operation at most k times in order to reduce a given graph parameter π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document} by at least d? We show that this problem is NP-hard when the parameter is the independence number and the graph operation is vertex deletion or edge contraction, even for fixed d=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=1$$\\end{document} and when restricted to chordal graphs. We give a polynomial time algorithm for bipartite graphs when the operation is edge contraction, the parameter is the independence number and d is fixed. Further, we complete the complexity dichotomy for H-free graphs when the parameter is the clique number and the operation is edge contraction by showing that this problem is NP-hard in (C3+P1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(C_3+P_1)$$\\end{document}-free graphs even for fixed d=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=1$$\\end{document}. When the operation is edge deletion and the parameter is the chromatic number, we determine the computational complexity of the associated problem for cographs and complete multipartite graphs. Our results answer several open questions stated in Diner et al. (Theor Comput Sci 746:49–72, 2012, https://doi.org/10.1016/j.tcs.2018.06.023).

A note on the connected game coloring number

We consider the \emph{connected game coloring number} of a graph, introduced by Charpentier et al. as a game theoretic graph parameter that measures the degeneracy of a graph with respect to a certain two-player game played with an uncooperative adversary. We consider the connected game coloring number of graphs of bounded treedepth and of $k$-trees. In particular, we show that there exists an outerplanar $2$-tree with connected game coloring number of $5$, which answers a question from [C. Charpentier, H. Hocquard, E. Sopena, and X. Zhu. A connected version of the graph coloring game. \textit{Discrete Appl. Math.}, 2020].

Transcranial Direct Current Stimulation Reverses Stroke-Induced Network Alterations in Mice.

Beyond focal effects, stroke lesions impact the function of distributed networks. We here investigated (1) whether transcranial direct current stimulation (tDCS) alters the network changes induced by cerebral ischemia and (2) whether functional network parameters predict the therapeutic efficacy of tDCS in a mouse model of focal photothrombotic stroke. Starting 3 days after stroke, cathodal tDCS (charge density=39.6 kC/m²) was applied over 10 days in male C57Bl/6J mice under light anesthesia over the lesioned sensory-motor cortex. Functional connectivity (resting-state functional magnetic resonance imaging) was evaluated for up to 28-day poststroke, with global graph parameters of network integration computed. Ischemia induced a subacute increase in connectivity accompanied by a significant reduction in characteristic path length, reversed by 10 days of tDCS. Early measures of functional network alterations and the network configuration at prestroke baseline predicted spontaneous and tDCS-augmented motor recovery. Stroke induces characteristic network changes throughout the brain that can be detected by resting-state functional magnetic resonance imaging. These network changes were, at least in part, reversed by tDCS. Moreover, early markers of a network impairment and the network configuration before the insult improve the prediction of motor recovery.

Graph parameters, implicit representations and factorial properties

How to efficiently represent a graph in computer memory is a fundamental data structuring question. In the present paper, we address this question from a combinatorial point of view. A representation of an n-vertex graph G is called implicit if it assigns to each vertex of G a binary code of length O(log⁡n) so that the adjacency of two vertices is a function of their codes. A necessary condition for a hereditary class X of graphs to admit an implicit representation is that X has at most factorial speed of growth. This condition, however, is not sufficient, as was recently shown in [19]. Several sufficient conditions for the existence of implicit representations deal with boundedness of some parameters, such as degeneracy or clique-width. In the present paper, we analyze more graph parameters and prove a number of new results related to implicit representation and factorial properties.

Double domination in rooted product graphs

A set D of vertices of a graph G is a double dominating set of G if |N[v]∩D|≥2 for every v∈V(G), where N[v] represents the closed neighbourhood of v. The double domination number of G is the minimum cardinality among all double dominating sets of G. In this article, we show that if G and H are graphs with no isolated vertex, then for any vertex v∈V(H) there are six possible expressions, in terms of domination parameters of the factor graphs, for the double domination number of the rooted product graph G∘vH. Additionally, we characterize the graphs G and H that satisfy each of these expressions.

Parameterized Algorithms for Colored Clustering

In the Colored Clustering problem, one is asked to cluster edge-colored (hyper-)graphs whose colors represent interaction types. More specifically, the goal is to select as many edges as possible without choosing two edges that share an endpoint and are colored differently. Equivalently, the goal can also be described as assigning colors to the vertices in a way that fits the edge-coloring as well as possible. As this problem is NP-hard, we build on previous work by studying its parameterized complexity. We give a 2ᴼ⁽ᵏ⁾·nᴼ⁽¹⁾-time algorithm where k is the number of edges to be selected and n the number of vertices. We also prove the existence of a problem kernel of size O(k⁵ᐟ²), resolving an open problem posed in the literature. We consider parameters that are smaller than k, the number of edges to be selected, and r, the number of edges that can be deleted. Such smaller parameters are obtained by considering the difference between k or r and some lower bound on these values. We give both algorithms and lower bounds for Colored Clustering with such parameterizations. Finally, we settle the parameterized complexity of Colored Clustering with respect to structural graph parameters by showing that it is W[1]-hard with respect to both vertex cover number and tree-cut width, but fixed-parameter tractable with respect to local feedback edge number.

Hat Guessing Numbers of Strongly Degenerate Graphs

Assume players are placed on vertices of a graph . The following game was introduced by Winkler: An adversary puts a hat on each player, where each hat has a color out of available colors. The players can see the hat of each of their neighbors in but cannot see their own hats. Using a predetermined guessing strategy, the players then simultaneously guess the color of their hats. The players win if at least one of them guesses correctly; otherwise, the adversary wins. The largest integer such that there is a winning strategy for the players is denoted by , and this is called the hat guessing number of . Although this game has received much attention in recent years, not much is known about how the hat guessing number relates to other graph parameters. For instance, a natural open question is whether the hat guessing number can be bounded from above in terms of degeneracy. In this paper, we prove that the hat guessing number of a graph can be bounded from above in terms of a related notion, which we call strong degeneracy. We further give an exact characterization of graphs with bounded strong degeneracy. As a consequence, we significantly improve the best known upper bound on the hat guessing number of outerplanar graphs from to 40 and further derive upper bounds on the hat guessing number for any class of -free graphs with bounded expansion, such as the class of -free planar graphs; more generally, for -free graphs with bounded Hadwiger number or without a -subdivision; and for Erdős–Rényi random graphs with constant average degree.

Asymptotic enumeration of graphs by degree sequence, and the degree sequence of a random graph

In this paper we relate a fundamental parameter of a random graph, its degree sequence, to a simple model of nearly independent binomial random variables. As a result, many interesting functions of the joint distribution of graph degrees, such as the distribution of the median degree, become amenable to estimation. Our result is established by proving an asymptotic formula conjectured in 1990 for the number of graphs with given degree sequence. In particular, this gives an asymptotic formula for the number of d -regular graphs for all d , as n\to\infty . The key to our results is a new approach to estimating ratios between point probabilities in the space of degree sequences of the random graph, including analysis of fixed points of the associated operators.

The micro and macro interactions in acute autoimmune encephalitis: a study of resting-state EEG.

Early recognition of autoimmune encephalitis (AIE) is often difficult and time-consuming. Understanding how the micro-level (antibodies) and macro-level (EEG) couple with each other may help rapidly diagnose and appropriately treat AIE. However, limited studies focused on brain oscillations involving micro- and macro-interactions in AIE from a neuro-electrophysiological perspective. Here, we investigated brain network oscillations in AIE using Graph theoretical analysis of resting state EEG. AIE Patients (n = 67) were enrolled from June 2018 to June 2022. Each participant underwent a ca.2-hour 19-channel EEG examination. Five 10-second resting state EEG epochs with eyes closed were extracted for each participant. The functional networks based on the channels and Graph theory analysis were carried out. Compared with the HC group, significantly decreased FC across whole brain regions at alpha and beta bands were found in AIE patients. In addition, the local efficiency and clustering coefficient of the delta band was higher in AIE patients than in the HC group (P < 0.05). AIE patients had a smaller world index (P < 0.05) and higher shortest path length (P < 0.001) in the alpha band than those of the control group. Also, the AIE patients' global efficiency, local efficiency, and clustering coefficients decreased in the alpha band (P < 0.001). Different types of antibodies (antibodies against ion channels, antibodies against synaptic excitatory receptors, antibodies against synaptic inhibitory receptors, and multiple antibodies positive) showed distinct graph parameters. Moreover, the graph parameters differed in the subgroups by intracranial pressure. Correlation analysis revealed that magnetic resonance imaging abnormalities were related to global efficiency, local efficiency, and clustering coefficients in the theta, alpha, and beta bands, but negatively related to the shortest path length. These findings add to our understanding of how brain FC and graph parameters change and how the micro- (antibodies) scales interact with the macro- (scalp EEG) scale in acute AIE. The clinical traits and subtypes of AIE may be suggested by graph properties. Further longitudinal cohort studies are needed to explore the associations between these graph parameters and recovery status, and their possible applications in AIE rehabilitation.

Edge and Pair Queries-Random Graphs and Complexity

We investigate two types of query games played on a graph, pair queries and edge queries. We concentrate on investigating the two associated graph parameters for binomial random graphs, and showing that determining any of the two parameters is NP-hard for bounded degree graphs.