A complete subgraph of a given graph is called a clique. A clique Polynomial of a graph is a generating function of the number of cliques in $G$. A real root of the clique polynomial of a graph $G$ is called a \emph{clique root} of $G$. \\ Hajiabolhassan and Mehrabadi showed that the clique polynomial of any simple graph has a clique root in $[-1,0)$. As a generalization of their result, the author of this paper showed that the class of $K_{4}$-free connected chordal graphs has also only clique roots. \\ A given graph $G$ is called flat if each edge of $G$ belongs to at most two triangles of $G$. In answering the author's open question about the class of \emph{non-chordal} graphs with the same property of having only c;ique roots, we extend the aforementioned result to the class of $K_{4}$-free flat graphs. In particular, we prove that the class of $K_{4}$-free flat graphs without isolated edges has $r=-1$ as one of its clique roots. We finally present some interesting open questions and conjectures regarding clique roots of graphs.
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A famous result by Erd\H{o}s and Szekeres (1935) asserts that, for all $k,d \in \mathbb{N}$, there is a smallest integer $n = g^{(d)}(k)$ such that every set of at least $n$ points in $\mathbb{R}^d$ in general position contains a $k$-gon, that is, a subset of $k$ points which is in convex position. In this article, we present a SAT model based on acyclic chirotopes (oriented matroids) to investigate Erd\H{o}s--Szekeres numbers in small dimensions. To solve the SAT instances we use modern SAT solvers and all our unsatisfiability results are verified using DRAT certificates. We show $g^{(3)}(7) = 13$, $g^{(4)}(8) \le 13$, and $g^{(5)}(9) \le 13$, which are the first improvements for decades. For the setting of $k$-holes (i.e., $k$-gons with no other points in the convex hull), where $h^{(d)}(k)$ denotes the minimum number $n$ such that every set of at least $n$ points in $\mathbb{R}^d$ in general position contains a $k$-hole, we show $h^{(3)}(7) \le 14$, $h^{(4)}(8) \le 13$, and $h^{(5)}(9) \le 13$. Moreover, all obtained bounds are sharp in the setting of acyclic chirotopes and we conjecture them to be sharp also in the original setting of point sets. As a byproduct, we verify previously known bounds. In particular, we present the first computer-assisted proof of the upper bound $h^{(2)}(6)\le g^{(2)}(9) \le 1717$ by Gerken (2008).
We study the relationship between the eternal domination number of a graph and its clique covering number using both large-scale computation and analytic methods. In doing so, we answer two open questions of Klostermeyer and Mynhardt. We show that the smallest graph having its eternal domination number less than its clique covering number has $10$ vertices. We determine the complete set of $10$-vertex and $11$-vertex graphs having eternal domination numbers less than their clique covering numbers. We show that the smallest triangle-free graph with this property has order $13$, as does the smallest circulant graph. We describe a method to generate an infinite family of triangle-free graphs and an infinite family of circulant graphs with eternal domination numbers less than their clique covering numbers. We also consider planar graphs and cubic graphs. Finally, we show that for any integer $k \geq 2$ there exist infinitely many graphs having domination number and eternal domination number equal to $k$ containing dominating sets which are not eternal dominating sets.
This paper studies a Stackelberg game wherein a sender (leader) attempts to shape the information of a less informed receiver (follower) who in turn takes an action that determines the payoff of both players. The sender chooses signals to maximize its own utility function while the receiver aims to ascertain the value of a source that is privately known to the sender. It is well known that such sender-receiver games admit a vast number of equilibria and not all signals from the sender can be relied on as truthful. Our main contribution is an exact characterization of the minimum number of distinct source symbols that can be correctly recovered by a receiver in \textit{any} equilibrium of this game; we call this quantity the \textit{informativeness} of the sender. We show that the informativeness is given by the \textit{vertex clique cover number} of a certain graph induced by the utility function, whereby it can be computed based on the utility function alone without the need to enumerate all equilibria. We find that informativeness characterizes the existence of well-known classes of separating, pooling and semi-separating equilibria. We also compare informativeness with the amount of information obtained by the receiver when it is the leader and show that the informativeness is always greater than the latter, implying that the receiver is better off being a follower.
We consider a core language of graph queries. These queries are seen as formulas to be solved with respect to graph-oriented databases. For this purpose, we first define a graph query algebra where some operations over graphs and sets of graph homomorphisms are specified. Then, the notion of pattern is introduced to represent a kind of recursively defined formula over graphs. The syntax and formal semantics of patterns are provided. Afterwards, we propose a new sound and complete calculus to solve patterns. This calculus, which is based on a rewriting system, develops only one derivation per pattern to be solved. Our calculus is generic in the sense that it can be adapted to different kinds of graph databases provided that the notions of graph and graph homomorphism (match) are well defined.
We show that CC-circuits of bounded depth have the same expressive power as polynomials over finite nilpotent algebras from congruence modular varieties. We use this result to phrase and discuss an algebraic version of Barrington, Straubing and Th\'erien's conjecture, which states that CC-circuits of bounded depth need exponential size to compute AND. Furthermore we investigate the complexity of deciding identities and solving equations in a fixed nilpotent algebra. Under the assumption that the conjecture is true, we obtain quasipolynomial algorithms for both problems. On the other hand, if AND is computable by uniform CC-circuits of bounded depth and polynomial size, we can construct a nilpotent algebra with coNP-complete, respectively NP-complete problem.
We solve the problem of existence of perfect codes in the Doob graph. It is shown that 1-perfect codes in the Doob graph D(m,n) exist if and only if 6m+3n+1 is a power of 2; that is, if the size of a 1-ball divides the number of vertices. Keywords: perfect codes, distance-regular graphs, Doob graphs, Eisenstein-Jacobi integers.
A homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$. Let $H$ be a fixed graph with possible loops. In the list homomorphism problem, denoted by \textsc{LHom}($H$), the instance is a graph $G$, whose every vertex is equipped with a subset of $V(H)$, called list. We ask whether there exists a homomorphism from $G$ to $H$, such that every vertex from $G$ is mapped to a vertex from its list. We study the complexity of the \textsc{LHom}($H$) problem in intersection graphs of various geometric objects. In particular, we are interested in answering the question for what graphs $H$ and for what types of geometric objects, the \textsc{LHom}($H$) problem can be solved in time subexponential in the number of vertices of the instance. We fully resolve this question for string graphs, i.e., intersection graphs of continuous curves in the plane. Quite surprisingly, it turns out that the dichotomy exactly coincides with the analogous dichotomy for graphs excluding a fixed path as an induced subgraph [Okrasa, Rz\k{a}\.zewski, STACS 2021]. Then we turn our attention to subclasses of string graphs, defined as intersections of fat objects. We observe that the (non)existence of subexponential-time algorithms in such classes is closely related to the size $\mathrm{mrc}(H)$ of a maximum reflexive clique in $H$, i.e., maximum number of pairwise adjacent vertices, each of which has a loop. We study the maximum value of $\mathrm{mrc}(H)$ that guarantees the existence of a subexponential-time algorithm for \textsc{LHom}($H$) in intersection graphs of (i) convex fat objects, (ii) fat similarly-sized objects, and (iii) disks. In the first two cases we obtain optimal results, by giving matching algorithms and lower bounds. Finally, we discuss possible extensions of our results to weighted generalizations of \textsc{LHom}($H$).
Given the input graph and its label/property, several key problems of graph learning, such as finding interpretable subgraphs, graph denoising and graph compression, can be attributed to the fundamental problem of recognizing a subgraph of the original one. This subgraph shall be as informative as possible, yet contains less redundant and noisy structure. This problem setting is closely related to the well-known information bottleneck (IB) principle, which, however, has less been studied for the irregular graph data and graph neural networks (GNNs). In this paper, we propose a framework of Graph Information Bottleneck (GIB) for the subgraph recognition problem in deep graph learning. Under this framework, one can recognize the maximally informative yet compressive subgraph, named IB-subgraph. However, the GIB objective is notoriously hard to optimize, mostly due to the intractability of the mutual information of irregular graph data and the unstable optimization process. In order to tackle these challenges, we propose: i) a GIB objective based-on a mutual information estimator for the irregular graph data; ii) a bi-level optimization scheme to maximize the GIB objective; iii) a connectivity loss to stabilize the optimization process. We evaluate the properties of the IB-subgraph in three application scenarios: improvement of graph classification, graph interpretation and graph denoising. Extensive experiments demonstrate that the information-theoretic IB-subgraph enjoys superior graph properties.
The focus of Part I of this monograph has been on both the fundamental properties, graph topologies, and spectral representations of graphs. Part II embarks on these concepts to address the algorithmic and practical issues centered round data/signal processing on graphs, that is, the focus is on the analysis and estimation of both deterministic and random data on graphs. The fundamental ideas related to graph signals are introduced through a simple and intuitive, yet illustrative and general enough case study of multisensor temperature field estimation. The concept of systems on graph is defined using graph signal shift operators, which generalize the corresponding principles from traditional learning systems. At the core of the spectral domain representation of graph signals and systems is the Graph Discrete Fourier Transform (GDFT). The spectral domain representations are then used as the basis to introduce graph signal filtering concepts and address their design, including Chebyshev polynomial approximation series. Ideas related to the sampling of graph signals are presented and further linked with compressive sensing. Localized graph signal analysis in the joint vertex-spectral domain is referred to as the vertex-frequency analysis, since it can be considered as an extension of classical time-frequency analysis to the graph domain of a signal. Important topics related to the local graph Fourier transform (LGFT) are covered, together with its various forms including the graph spectral and vertex domain windows and the inversion conditions and relations. A link between the LGFT with spectral varying window and the spectral graph wavelet transform (SGWT) is also established. Realizations of the LGFT and SGWT using polynomial (Chebyshev) approximations of the spectral functions are further considered. Finally, energy versions of the vertex-frequency representations are introduced.
Many problems in areas as diverse as recommendation systems, social network analysis, semantic search, and distributed root cause analysis can be modeled as pattern search on labeled graphs (also called "heterogeneous information networks" or HINs). Given a large graph and a query pattern with node and edge label constraints, a fundamental challenge is to nd the top-k matches ac- cording to a ranking function over edge and node weights. For users, it is di cult to select value k . We therefore propose the novel notion of an any-k ranking algorithm: for a given time budget, re- turn as many of the top-ranked results as possible. Then, given additional time, produce the next lower-ranked results quickly as well. It can be stopped anytime, but may have to continues until all results are returned. This paper focuses on acyclic patterns over arbitrary labeled graphs. We are interested in practical algorithms that effectively exploit (1) properties of heterogeneous networks, in particular selective constraints on labels, and (2) that the users often explore only a fraction of the top-ranked results. Our solution, KARPET, carefully integrates aggressive pruning that leverages the acyclic nature of the query, and incremental guided search. It enables us to prove strong non-trivial time and space guarantees, which is generally considered very hard for this type of graph search problem. Through experimental studies we show that KARPET achieves running times in the order of milliseconds for tree patterns on large networks with millions of nodes and edges.
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