914 research outputs found
Shallow Minors, Graph Products and Beyond Planar Graphs
The planar graph product structure theorem of Dujmovi\'{c}, Joret, Micek,
Morin, Ueckerdt, and Wood [J. ACM 2020] states that every planar graph is a
subgraph of the strong product of a graph with bounded treewidth and a path.
This result has been the key tool to resolve important open problems regarding
queue layouts, nonrepetitive colourings, centered colourings, and adjacency
labelling schemes. In this paper, we extend this line of research by utilizing
shallow minors to prove analogous product structure theorems for several beyond
planar graph classes. The key observation that drives our work is that many
beyond planar graphs can be described as a shallow minor of the strong product
of a planar graph with a small complete graph. In particular, we show that
powers of planar graphs, -planar, -cluster planar, fan-planar and
-fan-bundle planar graphs have such a shallow-minor structure. Using a
combination of old and new results, we deduce that these classes have bounded
queue-number, bounded nonrepetitive chromatic number, polynomial -centred
chromatic numbers, linear strong colouring numbers, and cubic weak colouring
numbers. In addition, we show that -gap planar graphs have at least
exponential local treewidth and, as a consequence, cannot be described as a
subgraph of the strong product of a graph with bounded treewidth and a path
Sparse universal graphs for planarity
We show that for every integer there exists a graph with
vertices and edges such that every -vertex planar
graph is isomorphic to a subgraph of . The best previous bound on the
number of edges was , proved by Babai, Chung, Erd\H{o}s, Graham,
and Spencer in 1982. We then show that for every integer there is a
graph with vertices and edges that contains induced copies
of every -vertex planar graph. This significantly reduces the number of
edges in a recent construction of the authors with Dujmovi\'c, Gavoille, and
Micek.Comment: v2: added new result about induced-universal graph
Graph Product Structure for h-Framed Graphs
Graph product structure theory expresses certain graphs as subgraphs of the strong product of much simpler graphs. In particular, an elegant formulation for the corresponding structural theorems involves the strong product of a path and of a bounded treewidth graph, and allows to lift combinatorial results for bounded treewidth graphs to graph classes for which the product structure holds, such as to planar graphs [Dujmovi? et al., J. ACM, 67(4), 22:1-38, 2020].
In this paper, we join the search for extensions of this powerful tool beyond planarity by considering the h-framed graphs, a graph class that includes 1-planar, optimal 2-planar, and k-map graphs (for appropriate values of h). We establish a graph product structure theorem for h-framed graphs stating that the graphs in this class are subgraphs of the strong product of a path, of a planar graph of treewidth at most 3, and of a clique of size 3? h/2 ?+? h/3 ?-1. This allows us to improve over the previous structural theorems for 1-planar and k-map graphs. Our results constitute significant progress over the previous bounds on the queue number, non-repetitive chromatic number, and p-centered chromatic number of these graph classes, e.g., we lower the currently best upper bound on the queue number of 1-planar graphs and k-map graphs from 115 to 82 and from ? 33/2(k+3 ? k/2? -3)? to ? 33/2 (3? k/2 ?+? k/3 ?-1) ?, respectively. We also employ the product structure machinery to improve the current upper bounds on the twin-width of 1-planar graphs from O(1) to 80. All our structural results are constructive and yield efficient algorithms to obtain the corresponding decompositions
The Ising model and beyond
We study the SU(3) AVE graphs, which appear in the classification of modular in variant partition functions from numerous viewpoints, including determination of their Boltzmann weights, representations of Hecke algebras, a new notion of A2 planar algebras and their modules, various Hilbert series of dimensions and spectral measures, and the K-theory of associated Cuntz-Krieger algebras. We compute the K-theory of the of the Cuntz-Krieger algebras associated to the SU(3) AVE graphs. We compute the numerical values of the Ocneanu cells, and consequently representations of the Hecke algebra, for the AVE graphs. Some such representations have appeared in the literature and we compare our results. We use these cells to define an SU(3) analogue of the Goodman-de la Harpe-Jones construction of a subfactor, where we embed the j42-Temperley-Lieb algebra in an AF path-algebra of the SU(3) AVE graphs. Using this construction, we realize all SU(3) modular invariants by subfactors previously announced by Ocneanu. We give a diagrammatic representation of the i42-Temperley-Lieb algebra, and show that it is isomorphic to Wenzl's representation of a Hecke algebra. Generalizing Jones's notion of a planar algebra, we construct an 42-planar algebra which captures the structure contained in the SU(3) AVE subfactors. We show that the subfactor for an AVE graph with a flat connection has a description as a flat >12-planar algebra. We introduce the notion of modules over an 42-planar algebra, and describe certain irreducible Hilbert A2- Temperley-Lieb-modules. A partial decomposition of the ,42-planar algebras for the AVE graphs is achieved. We compare various Hilbert series of dimensions associated to ADE models for SU(2), and the Hilbert series of certain Calabi-Yau algebras of dimension 3. We also consider spectral measures for the ADE graphs and generalize to SU(3), and in particular obtain spectral measures for the infinite SU(3) graphs
The Realizability of Theta Graphs as Reconfiguration Graphs of Minimum Independent Dominating Sets
The independent domination number of a graph is the minimum
cardinality of a maximal independent set of , also called an -set. The
-graph of is the graph whose vertices correspond to the -sets, and
where two -sets are adjacent if and only if they differ by two adjacent
vertices. Not all graphs are -graph realizable, that is, given a target
graph , there does not necessarily exist a source graph such that is
the -graph of . We consider a class of graphs called "theta graphs": a
theta graph is the union of three internally disjoint nontrivial paths with the
same two distinct end vertices. We characterize theta graphs that are -graph
realizable, showing that there are only finitely many that are not. We also
characterize those line graphs and claw-free graphs that are -graphs, and
show that all -connected cubic bipartite planar graphs are -graphs.Comment: 34 pages, 29 figure
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