Compact labelings for efficient first-order model-checking

Bipartite bithreshold graphs

An algorithm for finding clique cut-sets

We prove that there exists an O(log(n))-labeling scheme for every first-order formula with free set variables in every class of graphs that is nicely locally cwd-decomposable, which contains in particular, the nicely locally tree-decomposable classes. For every class of graphs of bounded expansionwe prove that every bounded formulahas an O(log(n))-labeling scheme. We also prove that every quantifier-free formula has an O(log(n))-labeling scheme in graphs of bounded arboricity. Some of these results are extended to counting queries.

In this paper, we define the distance-degree energy of a graph which provides the most natural generalization of energy, distance energy and Harary energy. Also, we derive certain properties of the distance-degree energy and investigate its behaviour on certain classes of graphs. Further, we study its relationship with other graph invariants and obtain a lower bound.

Zero forcing number of a graph is the minimum cardinality of the zero forcing set. A zero forcing set is a set of black vertices of minimum cardinality that can colour the entire graph black using the color change rule: each vertex of G is coloured either white or black, and vertex v is a black vertex and can force a white neighbour only if it has one white neighbour. In this paper we identify a class of graph where the zero forcing number is equal to the minimum rank of the graph and call it as a new class of graph that is open global shadow graph”. Some of the basic properties of open global shadow graph are studied. The zero forcing number of open global shadow graph of a graph with upper and lower bound is obtained. Hence giving the upper and lower bound for the minimum rank of the graph.

Numerous studies based on mathematical models and tools indicate that there is a strong inherent relationship between the chemical properties of the chemical compounds and drugs with their molecular structures. In the last two decades, the graph-theoretic techniques are frequently used to analyse the various physicochemical and structural properties of the molecular graphs which play a vital role in chemical engineering and pharmaceutical industry. In this paper, we compute Zagreb indices of the generalized sum graphs in the form of the different indices of their factor graphs, where generalized sum graphs are obtained under the operations of subdivision and strong product of graphs. Moreover, the obtained results are illustrated with the help of particular classes of graphs and analysed to find the efficient subclass with dominant indices.

This paper examines the problem of determining the threshold dimension of a graph. There exists numerous characterizations of threshold graphs, those graphs of threshold dimension one, as well as fast polynomial time algorithms to test if a graph is a threshold graph. Yannakakis [1982] proved that, in general, determining the threshold dimension is a hard problem by proving that for fixed $k\geqq 3$, determining if the threshold dimension of a graph is less than or equal to k is an NP-complete problem. In this paper we compute the threshold dimension of several classes of graphs and obtain upper and lower bounds on the threshold dimension of a graph. Counterexamples to a conjecture on threshold dimension are provided.

Hierarchical Glocal Attention Pooling for Graph Classification

This work studies decision problems from the perspective of nondeterministic distributed algorithms. For a yes instance there must exist a proof that can be verified with a distributed algorithm: all nodes must accept a valid proof, and at least one node must reject an invalid proof. We focus on locally checkable proofs that can be verified with a constant-time distributed algorithm.For example, it is easy to prove that a graph is bipartite: the locally checkable proof gives a 2-colouring of the graph, which only takes 1 bit per node. However, it is more difficult to prove that a graph is not bipartite - it turns out that any locally checkable proof requires ©(log n) bits per node.In this work we classify graph problems according to their local proof complexity, i.e., how many bits per node are needed in a locally checkable proof. We establish tight or near-tight results for classical graph properties such as the chromatic number. We show that the proof complexities form a natural hierarchy of complexity classes: for many classical graph problems, the proof complexity is either 0, (1), (log n), or poly(n) bits per node. Among the most difficult graph properties are symmetric graphs, which require ©(n²) bits per node, and non-3-colourable graphs, which require ©(n²/log n) bits per node - any pure graph property admits a trivial proof of size O(n²).

This paper illustrates the application of graph invariants to break symmetries for graph search problems. The paper makes two contributions: (1) the use of higher dimensional graph invariants in symmetry breaking constraints; and (2) a novel technique to obtain symmetry breaking constraints by combining graph invariants. Experimentation demonstrates that the proposed approach applies to provide new results for the generation of a particular class of cubic graphs.

Gauss diagram formulas of Vassiliev invariants of 2-bouquet graphs

Over the last decade, first-order constraints have been efficiently used in the artificial intelligence world to model many kinds of complex problems such as: scheduling, resource allocation, computer graphics and bio-informatics. Recently, a new property called decomposability has been introduced and many first-order theories have been proved to be decomposable: finite or infinite trees, rational and real numbers, linear dense order,...etc. A decision procedure in the form of five rewriting rules has also been developed. This latter can decide if a first-order formula without free variables is true or not in any decomposable theory. Unfortunately, this decision procedure is not enough when we want to express the solutions of a first-order constraint having free variables. These kind of problems are generally known as first-order constraint satisfaction problems. We present in this paper, not only a decision procedure but a full first-order constraint solver for decomposable theories. Our solver is given in the form of nine rewriting rules which transform any first-order constraint ? (which can possibly contain free variables) into an equivalent formula ? which is either the formula true, or the formula false or a simple solved formula having at least one free variable and being equivalent neither to true nor to false. We show the efficiency of our solver by solving complex first-order constraints over finite or infinite trees containing a huge number of imbricated quantifiers and negations and compare the performances with those obtained using the most recent and efficient dedicated solver for finite or infinite trees. This is the first full first-order constraint solver for any decomposable theory.

On edge transitivity of directed graphs

In this paper, we explain the regularity, projective dimension and depth of the edge ideal of some classes of graphs in terms of invariants of graphs. We show that for a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C 5"> <mml:semantics> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>5</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">C_5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -free vertex decomposable graph <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="reg left-parenthesis upper R slash upper I left-parenthesis upper G right-parenthesis right-parenthesis equals c Subscript upper G Baseline"> <mml:semantics> <mml:mrow> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mtext>reg</mml:mtext> </mml:mstyle> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>I</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mi>G</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mbox {reg}(R/I(G))= c_G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c Subscript upper G"> <mml:semantics> <mml:msub> <mml:mi>c</mml:mi> <mml:mi>G</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">c_G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the maximum number of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -disjoint edges in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Moreover, for this class of graphs we characterize <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pd left-parenthesis upper R slash upper I left-parenthesis upper G right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mtext>pd</mml:mtext> </mml:mstyle> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>I</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mbox {pd}(R/I(G))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="depth left-parenthesis upper R slash upper I left-parenthesis upper G right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mtext>depth</mml:mtext> </mml:mstyle> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>I</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mbox {depth}(R/I(G))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . As a corollary we describe these invariants in forests and sequentially Cohen-Macaulay bipartite graphs.

Addressable data graphs enjoy structural properties which are at once mathematically attractive and practically useful. This paper is devoted to studying generalizations of addressability, with a twofold goal. The first aim of the paper is to gain understanding of what features of addressing schemes account for the attractive properties of addressable data graphs. The second purpose of the paper is to seek a variant of the property of addressability, which retains the practical concomitants of the original property, but which is enjoyed by a broader class of data graphs. Two such variants are discovered,quasi-addressability andweak-addressability. Although “most” data graphs do not enjoy any form of addressability, every data graph can be slightly modified to render it quasi or weakly addressable—in sharp contrast to the stringent demands of addressability. However, the added breadth of these new addressabilities is obtained at the expense of the invariance and selectivity which accompany the original property.