2,978 research outputs found
Cohen-Macaulay binomial edge ideals of small graphs
A combinatorial property that characterizes Cohen-Macaulay binomial edge ideals has long been elusive. A recent conjecture ties the Cohen-Macaulayness of a binomial edge ideal JG to special disconnecting sets of vertices of its underlying graph G, called cut sets. More precisely, the conjecture states that JG is Cohen-Macaulay if and only if JG is unmixed and the collection of the cut sets of G is an accessible set system. In this paper we prove the conjecture theoretically for all graphs with up to 12 vertices and develop an algorithm that allows to computationally check the conjecture for all graphs with up to 15 vertices and all blocks with whiskers where the block has at most 11 vertices. This significantly extends previous computational results
A restricted L(2, 1)-labelling problem on interval graphs
In a graph G = (V, E), L(2, 1)-labelling is considered by a function ` whose domain is V and codomain is set of non-negative integers with a condition that the vertices which are adjacent assign labels whose difference is at least two and the vertices whose distance is two, assign distinct labels. The difference between maximum and minimum labels among all possible labels is denoted by λ2,1(G). This paper contains a variant of L(2, 1)-labelling problem. In L(2, 1)-labelling problem, all the vertices are L(2, 1)-labeled by least number of labels. In this paper, maximum allowable label K is given. The problem is: L(2, 1)-label the vertices of G by using the labels {0, 1, 2, . . . , K} such that maximum number of vertices get label. If K labels are adequate for labelling all the vertices of the graph then all vertices get label, otherwise some vertices remains unlabeled. An algorithm is designed to solve this problem. The algorithm is also illustrated by examples. Also, an algorithm is designed to test whether an interval graph is no hole label or not for the purpose of L(2, 1)-labelling.Publisher's Versio
On the uniqueness of collections of pennies and marbles
In this note we study the uniqueness problem for collections of pennies and
marbles. More generally, consider a collection of unit -spheres that may
touch but not overlap. Given the existence of such a collection, one may
analyse the contact graph of the collection. In particular we consider the
uniqueness of the collection arising from the contact graph. Using the language
of graph rigidity theory, we prove a precise characterisation of uniqueness
(global rigidity) in dimensions 2 and 3 when the contact graph is additionally
chordal. We then illustrate a wide range of examples in these cases. That is,
we illustrate collections of marbles and pennies that can be perturbed
continuously (flexible), are locally unique (rigid) and are unique (globally
rigid). We also contrast these examples with the usual generic setting of graph
rigidity.Comment: 9 pages, 11 figure
Homological Neural Networks: A Sparse Architecture for Multivariate Complexity
The rapid progress of Artificial Intelligence research came with the
development of increasingly complex deep learning models, leading to growing
challenges in terms of computational complexity, energy efficiency and
interpretability. In this study, we apply advanced network-based information
filtering techniques to design a novel deep neural network unit characterized
by a sparse higher-order graphical architecture built over the homological
structure of underlying data. We demonstrate its effectiveness in two
application domains which are traditionally challenging for deep learning:
tabular data and time series regression problems. Results demonstrate the
advantages of this novel design which can tie or overcome the results of
state-of-the-art machine learning and deep learning models using only a
fraction of parameters
Product structure of graph classes with strongly sublinear separators
We investigate the product structure of hereditary graph classes admitting
strongly sublinear separators. We characterise such classes as subgraphs of the
strong product of a star and a complete graph of strongly sublinear size. In a
more precise result, we show that if any hereditary graph class
admits separators, then for any fixed
every -vertex graph in is a subgraph
of the strong product of a graph with bounded tree-depth and a complete
graph of size . This result holds with if
we allow to have tree-depth . Moreover, using extensions of
classical isoperimetric inequalties for grids graphs, we show the dependence on
in our results and the above bound are
both best possible. We prove that -vertex graphs of bounded treewidth are
subgraphs of the product of a graph with tree-depth and a complete graph of
size , which is best possible. Finally, we investigate the
conjecture that for any hereditary graph class that admits
separators, every -vertex graph in is a
subgraph of the strong product of a graph with bounded tree-width and a
complete graph of size . We prove this for various classes
of interest.Comment: v2: added bad news subsection; v3: removed section "Polynomial
Expansion Classes" which had an error, added section "Lower Bounds", and
added a new author; v4: minor revisions and corrections
Induced subgraphs of zero-divisor graphs
Funding: Peter J. Cameron acknowledges the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Groups, representations and applications: new perspectives (supported by EPSRC grant no. EP/R014604/1), where he held a Simons Fellowship. For this research, T. Kavaskar was supported by the University Grant Commissions Start-Up Grant, Government of India grant No. F. 30-464/2019 (BSR) dated 27.03. T. Tamizh Chelvam was supported by CSIR Emeritus Scientist Scheme (No. 21 (1123)/20/EMR-II) of Council of Scientific and Industrial Research, Government of India.The zero-divisor graph of a finite commutative ring with unity is the graph whose vertex set is the set of zero-divisors in the ring, with a and b adjacent if ab=0. We show that the class of zero-divisor graphs is universal, in the sense that every finite graph is isomorphic to an induced subgraph of a zero-divisor graph. This remains true for various restricted classes of rings, including boolean rings, products of fields, and local rings. But in more restricted classes, the zero-divisor graphs do not form a universal family. For example, the zero-divisor graph of a local ring whose maximal ideal is principal is a threshold graph; and every threshold graph is embeddable in the zero-divisor graph of such a ring. More generally, we give necessary and sufficient conditions on a non-local ring for which its zero-divisor graph to be a threshold graph. In addition, we show that there is a countable local ring whose zero-divisor graph embeds the Rado graph , and hence every finite or countable graph, as induced subgraph. Finally, we consider embeddings in related graphs such as the 2-dimensional dot product graph.Publisher PDFPeer reviewe
Topology of Cut Complexes of Graphs
We define the -cut complex of a graph with vertex set to be the
simplicial complex whose facets are the complements of sets of size in
inducing disconnected subgraphs of . This generalizes the Alexander
dual of a graph complex studied by Fr\"oberg (1990), and Eagon and Reiner
(1998). We describe the effect of various graph operations on the cut complex,
and study its shellability, homotopy type and homology for various families of
graphs, including trees, cycles, complete multipartite graphs, and the prism
, using techniques from algebraic topology, discrete Morse
theory and equivariant poset topology.Comment: 36 pages, 10 figures, 1 table, Extended Abstract accepted for
FPSAC2023 (Davis
Thick Forests
We consider classes of graphs, which we call thick graphs, that have their
vertices replaced by cliques and their edges replaced by bipartite graphs. In
particular, we consider the case of thick forests, which are a subclass of
perfect graphs. We show that this class can be recognised in polynomial time,
and examine the complexity of counting independent sets and colourings for
graphs in the class. We consider some extensions of our results to thick graphs
beyond thick forests.Comment: 40 pages, 19 figure
Clique‐width: Harnessing the power of atoms
Many NP-complete graph problems are polynomial-time solvable on graph classes of bounded clique-width. Several of these problems are polynomial-time solvable on a hereditary graph class if they are so on the atoms (graphs with no clique cut-set) of . Hence, we initiate a systematic study into boundedness of clique-width of atoms of hereditary graph classes. A graph is -free if is not an induced subgraph of , and it is -free if it is both -free and -free. A class of -free graphs has bounded clique-width if and only if its atoms have this property. This is no longer true for -free graphs, as evidenced by one known example. We prove the existence of another such pair and classify the boundedness of clique-width on -free atoms for all but 18 cases
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