Subgraph Of Graph Research Articles

Monochromatic subgraphs in graphs

Monochromatic subgraphs in graphs

Motif-Backdoor: Rethinking the Backdoor Attack on Graph Neural Networks via Motifs

Graph neural network (GNN) with a powerful representation capability has been widely applied to various areas. Recent works have exposed that GNN is vulnerable to the backdoor attack, i.e., models trained with maliciously crafted training samples are easily fooled by patched samples. Most of the proposed studies launch the backdoor attack using a trigger that is either the randomly generated subgraph e.g., erdős-rényi backdoor (ER-B) for less computational burden or the gradient-based generative subgraph e.g., graph trojaning attack (GTA) to enable a more effective attack. However, the interpretation of how is the trigger structure and the effect of the backdoor attack related has been overlooked in the current literature. Motifs, recurrent and statistically significant subgraphs in graphs, contain rich structure information. In this article, we are rethinking the trigger from the perspective of motifs and propose a motif-based backdoor attack, denoted as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Motif-Backdoor</i> . It contributes from three aspects. First, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Interpretation</i> , it provides an in-depth explanation for backdoor effectiveness by the validity of the trigger structure from motifs, leading to some novel insights, e.g., using subgraphs that appear less frequently in the graph as the trigger can achieve better attack performance. Second, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Effectiveness</i> , Motif-Backdoor reaches the state-of-the-art (SOTA) attack performance in both black-box and defensive scenarios. Third, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Efficiency</i> , based on the graph motif distribution, Motif-Backdoor can quickly obtain an effective trigger structure without target model feedback or subgraph model generation. Extensive experimental results show that Motif-Backdoor realizes the SOTA performance on three popular models and four public datasets compared with five baselines, e.g., Motif-Backdoor improves the attack success rate (ASR) by 14.73% compared with baselines on average. In addition, under a possible defense, Motif-Backdoor still implements satisfying performance, highlighting the requirement of defenses against backdoor attacks on GNNs. The datasets and code are available at https://github.com/Seaocn/Motif-Backdoor.

Removable and forced subgraphs of graphs

A connected graph G is H-removable if H is a subgraph of G, G has at least |V(H)|+2 vertices and G−V(H′) has a perfect matching for every subgraph H′ of G isomorphic to H. So a connected graph is kK2-removable if and only if it is k-extendable, where kK2 is a matching of size k. Further G is an H-forced graph if G−V(H′) has a unique perfect matching for every subgraph H′ of G isomorphic to H. In this paper, we first characterize kK2-forced graphs for k≥2 as K2k+2 and Kk+1,k+1 by using randomly matchable graphs. Let Pi denote a path with i vertices. We show that P3- and P4-removable graphs are factor-critical and 1-extendable respectively. By using ear decomposition, we obtain that a connected graph G is P3-forced if and only if G is K5 or C2k+1 (k≥2). Finally we prove that a connected graph G is P4-forced if and only if G is isomorphic to K6, K3,3, C2k (k≥3), the Petersen graph, a uniform theta graph Θ3n (n≥3) or C12+.

Grassmannians of codes

Consider the point line-geometry Pt(n,k) having as points all the [n,k]-linear codes having minimum dual distance at least t+1 and where two points X and Y are collinear whenever X∩Y is a [n,k−1]-linear code having minimum dual distance at least t+1. We are interested in the collinearity graph Λt(n,k) of Pt(n,k). The graph Λt(n,k) is a subgraph of the Grassmann graph and also a subgraph of the graph Δt(n,k) of the linear codes having minimum dual distance at least t+1 introduced in [9]. We shall study the structure of Λt(n,k) in relation to that of Δt(n,k) and we will characterize the set of its isolated vertices. We will then focus on Λ1(n,k) and Λ2(n,k) providing necessary and sufficient conditions for them to be connected.

Induced subgraphs and tree decompositions V. one neighbor in a hole

AbstractWhat are the unavoidable induced subgraphs of graphs with large treewidth? It is well‐known that the answer must include a complete graph, a complete bipartite graph, all subdivisions of a wall and line graphs of all subdivisions of a wall (we refer to these graphs as the “basic treewidth obstructions”). So it is natural to ask whether graphs excluding the basic treewidth obstructions as induced subgraphs have bounded treewidth. Sintiari and Trotignon answered this question in the negative. Their counterexamples, the so‐called “layered wheels,” contain wheels, where a wheel consists of a hole (i.e., an induced cycle of length at least four) along with a vertex with at least three neighbors in the hole. This leads one to ask whether graphs excluding wheels and the basic treewidth obstructions as induced subgraphs have bounded treewidth. This also turns out to be false due to Davies' recent example of graphs with large treewidth, no wheels and no basic treewidth obstructions as induced subgraphs. However, in Davies' example there exist holes and vertices (outside of the hole) with two neighbors in them. Here we prove that a hole with a vertex with at least two neighbors in it is inevitable in graphs with large treewidth and no basic obstruction. Our main result is that graphs in which every vertex has at most one neighbor in every hole (that does not contain it) and with the basic treewidth obstructions excluded as induced subgraphs have bounded treewidth.

Treewidth versus clique number. II. Tree-independence number

In 2020, we initiated a systematic study of graph classes in which the treewidth can only be large due to the presence of a large clique, which we call (tw,ω)-bounded. The family of (tw,ω)-bounded graph classes provides a unifying framework for a variety of very different families of graph classes, including graph classes of bounded treewidth, graph classes of bounded independence number, intersection graphs of connected subgraphs of graphs with bounded treewidth, and graphs in which all minimal separators are of bounded size. While Chaplick and Zeman showed in 2017 that (tw,ω)-bounded graph classes enjoy some good algorithmic properties related to clique and coloring problems, it is an interesting open problem to which extent (tw,ω)-boundedness has useful algorithmic implications for problems related to independent sets. We provide a partial answer to this question by identifying a sufficient condition for (tw,ω)-bounded graph classes to admit a polynomial-time algorithm for the Maximum Weight Independent Packing problem and, as a consequence, for the weighted variants of the Independent Set and Induced Matching problems.Our approach is based on a new min-max graph parameter related to tree decompositions. We define the independence number of a tree decomposition T of a graph as the maximum independence number over all subgraphs of G induced by some bag of T. The tree-independence number of a graph G is then defined as the minimum independence number over all tree decompositions of G. Boundedness of the tree-independence number is a refinement of (tw,ω)-boundedness that is still general enough to hold for all the aforementioned families of graph classes. Generalizing a result on chordal graphs due to Cameron and Hell from 2006, we show that if a graph is given together with a tree decomposition with bounded independence number, then the Maximum Weight Independent Packing problem can be solved in polynomial time. Applications of our general algorithmic result to specific graph classes are given in the third paper of the series [Dallard, Milanič, and Štorgel, Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure].

Induced subgraphs and tree decompositions II. Toward walls and their line graphs in graphs of bounded degree

This paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded maximum degree, asserting that for all k and Δ, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a subdivision of the (k×k)-wall or the line graph of a subdivision of the (k×k)-wall as an induced subgraph. We prove two theorems supporting this conjecture, as follows.1.For t≥2, a t-theta is a graph consisting of two nonadjacent vertices and three internally vertex-disjoint paths between them, each of length at least t. A t-pyramid is a graph consisting of a vertex v, a triangle B disjoint from v and three paths starting at v and vertex-disjoint otherwise, each joining v to a vertex of B, and each of length at least t. We prove that for all k,t and Δ, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a t-theta, or a t-pyramid, or the line graph of a subdivision of the (k×k)-wall as an induced subgraph. This affirmatively answers a question of Pilipczuk et al. asking whether every graph of bounded maximum degree and sufficiently large treewidth contains either a theta or a triangle as an induced subgraph (where a theta means a t-theta for some t≥2).2.A subcubic subdivided caterpillar is a tree of maximum degree at most three whose all vertices of degree three lie on a path. We prove that for every Δ and subcubic subdivided caterpillar T, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a subdivision of T or the line graph of a subdivision of T as an induced subgraph.

Rooted minors and locally spanning subgraphs

AbstractResults on the existence of various types of spanning subgraphs of graphs are milestones in structural graph theory and have been diversified in several directions. In the present paper, we consider “local” versions of such statements. In 1966, for instance, D. W. Barnette proved that a 3‐connected planar graph contains a spanning tree of maximum degree at most 3. A local translation of this statement is that if is a planar graph, is a subset of specified vertices of such that cannot be separated in by removing two or fewer vertices of , then has a tree of maximum degree at most 3 containing all vertices of . Our results constitute a general machinery for strengthening statements about ‐connected graphs (for ) to locally spanning versions, that is, subgraphs containing a set of a (not necessarily planar) graph in which only has high connectedness. Given a graph and , we say is a minor of rooted at , if is a minor of such that each bag of contains at most one vertex of and is a subset of the union of all bags. We show that has a highly connected minor rooted at if cannot be separated in by removing a few vertices of . Combining these investigations and the theory of Tutte paths in the planar case yields locally spanning versions of six well‐known results about degree‐bounded trees, Hamiltonian paths and cycles, and 2‐connected subgraphs of graphs.

Induced Subgraphs and Path Decompositions

A graph $H$ is an induced subgraph of a graph $G$ if a graph isomorphic to $H$ can be obtained from $G$ by deleting vertices. Recently, there has been significant interest in understanding the unavoidable induced subgraphs for graphs of large treewidth. Motivated by this work, we consider the analogous problem for pathwidth: what are the unavoidable induced subgraphs for graphs of large pathwidth? While resolving this question in the general setting looks challenging, we prove various results for sparse graphs. In particular, we show that every graph with bounded maximum degree and sufficiently large pathwidth contains a subdivision of a large complete binary tree or the line graph of a subdivision of a large complete binary tree as an induced subgraph. Similarly, we show that every graph excluding a fixed minor and with sufficiently large pathwidth contains a subdivision of a large complete binary tree or the line graph of a subdivision of a large complete binary tree as an induced subgraph. Finally, we present a characterisation for when a hereditary class defined by a finite set of forbidden induced subgraphs has bounded pathwidth.

On QSAR-based cardiotoxicity modeling with the expressiveness-enhanced graph learning model and dual-threshold scheme.

Introduction: Given the direct association with malignant ventricular arrhythmias, cardiotoxicity is a major concern in drug design. In the past decades, computational models based on the quantitative structure-activity relationship have been proposed to screen out cardiotoxic compounds and have shown promising results. The combination of molecular fingerprint and the machine learning model shows stable performance for a wide spectrum of problems; however, not long after the advent of the graph neural network (GNN) deep learning model and its variant (e.g., graph transformer), it has become the principal way of quantitative structure-activity relationship-based modeling for its high flexibility in feature extraction and decision rule generation. Despite all these progresses, the expressiveness (the ability of a program to identify non-isomorphic graph structures) of the GNN model is bounded by the WL isomorphism test, and a suitable thresholding scheme that relates directly to the sensitivity and credibility of a model is still an open question. Methods: In this research, we further improved the expressiveness of the GNN model by introducing the substructure-aware bias by the graph subgraph transformer network model. Moreover, to propose the most appropriate thresholding scheme, a comprehensive comparison of the thresholding schemes was conducted. Results: Based on these improvements, the best model attains performance with 90.4% precision, 90.4% recall, and 90.5% F1-score with a dual-threshold scheme (active: ; non-active: ). The improved pipeline (graph subgraph transformer network model and thresholding scheme) also shows its advantages in terms of the activity cliff problem and model interpretability.

Toughness and binding number bounds of star-like and path factor

Let $ \mathcal{L}$ be a set which consists of some connected graphs. LetEbe a spanning subgraph of graphG. It is called a $ \mathcal{L}$-factor if every component of it is isomorphic to the element in $ \mathcal{L}$. In this contribution, we give the lower bounds of four parameters (t(G), I(G), I′(G),bind (G)) ofG, which force the graphGadmits a ({K1,i:q ≤ i ≤ 2q − 1} ∪ {K2q+1})-factor forq ≥ 2 and a {P2, P2q+1}-factor forq ≥ 3 respectively. The tightness of the bounds are given.

Cheeger estimates of Dirichlet-to-Neumann operators on infinite subgraphs of graphs

In this paper, we study the Dirichlet-to-Neumann operators on infinite subgraphs of graphs. For an infinite subgraph, we prove Cheeger-type estimates for the bottom spectrum of the Dirichlet-to-Neumann operator, and the higher order Cheeger estimates for higher order eigenvalues of the Dirichlet-to-Neumann operator.

Long paths in heterogeneous random subgraphs of graphs with large minimum degree

For a graph Gk on n vertices with a minimum degree of at least k→∞ as n→∞, let G(n,p) be a random subgraph of Gk taken by retaining each edge (i,j) independently with probability pij and p={pij}(i,j)∈Gk. We show that under certain conditions on the edge probabilities, the resulting random graph has a long path that covers almost all or all vertices with probability tending to 1 as n→∞.

Novel FDP mechanisms for releasing bipartite graph data on fixed and infinite intervals

As data collection increases, more and more sensitive data is being used to publish query results. This creates a significant risk of privacy disclosure. As a mathematically provable privacy theory, differential privacy (DP) provides a tool to resist background knowledge attacks. Fuzzy differential privacy (FDP) generalizes differential privacy by employing smaller sensitivity and supporting multiple similarity measures. Thus the output error can be reduced under FDP. Existing FDP mechanisms employ sliding window strategy, which perturb the true query value to an interval with this value as the midpoint to maintain similarity of outputs from neighboring datasets. It is still possible for an attacker to infer some sensitive information based on the difference between the left and right endpoints of the output range. To address this issue, this article present two solutions: fixed interval perturbation and infinite interval perturbation. These strategies perturb the true query values of two neighboring datasets to the same interval and provide fuzzy differential privacy protection for the dataset. We apply the proposed method to the privacy-preserving problem of bipartite graph subgraph counting and verify the effectiveness by experiments.

Sober: Explores for invasive behaviour of malware

Trigger-based malware samples exhibit untrustworthy aims provided that certain environmental conditions are satisfied. We have categorized trigger-based malware behaviours into two classes, i.e., invasive and evasive. Invasive behaviour is concerned with malicious activities, while malware uses evasive behaviours for self-defence. In this article, we propose a greedy incremental approach for detecting invasive trigger-based malicious behaviour. The method proceeds by identifying and supplying resources required by malign samples. Trigger-based behaviours vary depending on environmental conditions. Such behaviours can be modelled as directed graphs, where each node represents a state, and edges denote invocation of a specific event. We define each state in the graph as atomic behaviour, represented using API calls, having legitimate functionality. This atomic behaviour transforms into harmful action if augmented with additional APIs. A well-known challenge with behavioural analysis is the proactive identification of malicious activities before devices are compromised. Timely prediction of malicious behaviours without damaging the systems can be fulfilled by defining them as atomic actions. To reckon a sub-behaviour as atomic, we propose a novel algorithm. Besides, a new likelihood statistical significance test, Htest, is suggested to extract frequent subgraphs of graphs, representing malware family signature. The main message underlying Htest is that the lesser the number of subgraphs exhibiting benign behaviour, the higher the chances of the sub-graph being chosen as malware family signature. Experiments conducted on 2320 real malware samples demonstrate a 42% reduction in the number of signatures than the state-of-the-art methods, namely CDG and CMQDG. Also, 3% and 1.3% improvement in path coverage compared to the state-of-the-art methods, namely Pytrigger and GoldenEye.

Some new results on sum index and difference index

&lt;abstract&gt;&lt;p&gt;Let $ G = (V(G), E(G)) $ be a graph with a vertex set $ V(G) $ and an edge set $ E(G) $. For every injective vertex labeling $ f:V\left (G \right)\to \mathbb{Z} $, there are two induced edge labelings denoted by $ f^{+} :E\left (G \right)\to \mathbb{Z} $ and $ f^{-} :E\left (G \right)\to \mathbb{Z} $. These two edge labelings $ f^{+} $ and $ f^{-} $ are defined by $ f^{+}(uv) = f(u)+f(v) $ and $ f^{-}(uv) = \left |f(u)-f(v)\right | $ for each $ uv\in E(G) $ with $ u, v\in V(G) $. The sum index and difference index of $ G $ are induced by the minimum ranges of $ f^{+} $ and $ f^{-} $, respectively. In this paper, we obtain the properties of sum and difference index labelings. We also improve the bounds on the sum indices and difference indices of regular graphs and induced subgraphs of graphs. Further, we determine the sum and difference indices of various families of graphs such as the necklace graphs and the complements of matchings, cycles and paths. Finally, we propose some conjectures and questions by comparison.&lt;/p&gt;&lt;/abstract&gt;

Structure of Projective Planar Subgraphs of the Graph Obstructions for Fixed Surface

Consider the problem of studying the metric properties of a subgraph G\v, where v is an arbitrary vertex of obstruction graphs G of a nonorientable genus, which will determine the sets of points of attachment of one subgraph to another and allow constructing prototypes of graphs-obstruction with number of vertices greater than 10 nonorientable genus greater than 1. This problem is related to Erdosh's hypothesis [3] on the coverage of obstruction graphs of an undirected surface of the genus k, where k &gt; 0, the smallest inclusion of the set of k + 1st graph of the homeomorphic K5, or K3,3, in [4] constructively proved for 35 minors of obstruction graphs of the projective plane, a set of 62 with no more than 10 vertices of obstruction graphs and their splits for the Klein surface, as well as some obstruction graphs for other surfaces. In [5], the existence of a finite set of obstruction graphs for a non-orienting surface was proved. A similar problem was considered in [6], where models or prototypes of obstruction graphs were considered. The prototype of the graph-obstruction of the undirected genus, we will call the graphs that have their own subgraph graph-obstruction of the undirected genus. In [7, 8] the tangent problem of covering the set of vertices with the smallest number of cycles-boundaries of 2-cells was considered, the concept of cell distance is given in [9, 10], where the boundaries of an oriented genus of graphs formed from planar graphs and a simple star glued to some of its peaks. Hypothetically, it is possible to obtain them by recursive φ-transformation of the graph-obstruction of the projective plane and a copy of its planar subgraph given on vertices, edges or parts of edges, or simple chains, i.e. achievable parts of the so-called graph-basis (graph of homeomorphic graph Kuratovsky plane). We assume that instead of one subgraph there can be several copies of subgraphs of graphs-obstructions of the projective plane.The article has an introduction and two parts, in which the structural properties of subgraphs of obstruction graphs for an undirected surface, presented as a φ-image of one of the Kuratovsky graphs and at least one planar graph, are investigated. The metric properties of the minimal embeddings of the subgraphs of the obstruction graphs for undirected surfaces are considered, and the main result is Theorems 1, 2, and Lemma 3 as the basis of the prototype construction algorithm. Keywords: φ-transformation of graphs, nonorientable surface, prototypes of graph-obstruction.

Irregular subgraphs

AbstractWe suggest two related conjectures dealing with the existence of spanning irregular subgraphs of graphs. The first asserts that any $d$ -regular graph on $n$ vertices contains a spanning subgraph in which the number of vertices of each degree between $0$ and $d$ deviates from $\frac{n}{d+1}$ by at most $2$ . The second is that every graph on $n$ vertices with minimum degree $\delta$ contains a spanning subgraph in which the number of vertices of each degree does not exceed $\frac{n}{\delta +1}+2$ . Both conjectures remain open, but we prove several asymptotic relaxations for graphs with a large number of vertices $n$ . In particular we show that if $d^3 \log n \leq o(n)$ then every $d$ -regular graph with $n$ vertices contains a spanning subgraph in which the number of vertices of each degree between $0$ and $d$ is $(1+o(1))\frac{n}{d+1}$ . We also prove that any graph with $n$ vertices and minimum degree $\delta$ contains a spanning subgraph in which no degree is repeated more than $(1+o(1))\frac{n}{\delta +1}+2$ times.

A counterexample to a conjecture about triangle-free induced subgraphs of graphs with large chromatic number

We prove that for every n, there is a graph G with χ(G)≥n and ω(G)≤3 such that every induced subgraph H of G with ω(H)≤2 satisfies χ(H)≤4.This disproves a well-known conjecture. Our construction is a digraph with bounded clique number, large dichromatic number, and no induced directed cycles of odd length at least 5.

Extremal functions for sparse minors

The notion of a graph minor, which generalizes graph subgraphs, is a central notion of modern graph theory. Classical results concerning graph minors include the Graph Minor Theorem and the Graph Structure Theorem, both due to Robertson and Seymour. The results concern properties of classes of graphs closed under taking minors; such graph classes include many important natural classes of graphs, e.g., the class of planar graphs and, more generally, the class of graphs embeddable in a fixed surface. The Graph Minor Theorem asserts that every class of graphs closed under taking minors has a finite list of forbidden minors. For example, Wagner’s Theorem, which claims that a graph is planar if and only if it does not contain or as a minor, is a particular case of this theorem. The Graph Structure Theorem asserts that graphs from a fixed class of graphs closed under taking minors can be decomposed in a tree-like fashion into graphs almost embeddable in a fixed surface. In particular, every graph in a class of graphs avoiding a fixed minor admits strongly sublinear separators (the Planar separator theorem of Lipton and Tarjan is a special case of this more general result). As the number of edges of every graph contained in a class of graphs closed under taking minors is linear in the number of its vertices, one can define to be the maximum possible density of a graph that does not contain a graph as a minor. This quantity has been a subject of very intensive research; for example, a long list of bounds concerning culminated with a result of Thomason in 2001, who precisely determined its asymptotic behavior. This paper provides bounds on when itself is from a class of sparse graphs. In particular, the authors prove an asymptotically tight bound on in terms of the number of vertices of and the ratio of the vertex cover and the number of vertices of graphs contained in a class of graphs with strongly sublinear separators.