Related papers: Planar graphs with separation are dp-minimal
A pseudoline arrangement graph is a planar graph induced by an embedding of a (simple) pseudoline arrangement. We study the corresponding graph realization problem and properties of pseudoline arrangement graphs. In the first part, we give…
Graph embedding is the major technique which is used to map guest graph into host graph. In architecture simulation, graph embedding is said to be one of the strongest application for the execution of parallel algorithm and simulation of…
We prove that any graph excluding $K_r$ as a minor has can be partitioned into clusters of diameter at most $\Delta$ while removing at most $O(r/\Delta)$ fraction of the edges. This improves over the results of Fakcharoenphol and Talwar,…
For a graph embedded into a surface, we relate many combinatorial parameters of the cycle matroid of the graph and the bond matroid of the dual graph with the topological parameters of the embedding. This will give an expression of the…
A planar 3-connected graph $G$ is called \emph{essentially $4$-connected} if, for every 3-separator $S$, at least one of the two components of $G-S$ is an isolated vertex. Jackson and Wormald proved that the length $\mathop{\rm…
We study entanglement properties of mixed density matrices obtained from combinatorial Laplacians. This is done by introducing the notion of the density matrix of a graph. We characterize the graphs with pure density matrices and show that…
The Bandwidth Theorem of B\"ottcher, Schacht and Taraz [Mathematische Annalen 343 (1), 175-205] gives minimum degree conditions for the containment of spanning graphs H with small bandwidth and bounded maximum degree. We generalise this…
We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several…
We establish that every embedding of a Grassmann graph in a polar Grassmann graph can be reduced to an embedding in a Grassmann graph or to an embedding in the collinearity graph of a polar space. Also, we consider $3$-embeddings, i.e.…
We show that for every $n\in\mathbb N$ and $\log n\le d\le n$, if a graph $G$ has $N=\Theta(dn)$ vertices and minimum degree $(1+o(1))\frac{N}{2}$, then it contains a spanning subdivision of every $n$-vertex $d$-regular graph.
A graph is circle if there is a family of chords in a circle such that two vertices are adjacent if the corresponding chords cross each other. There are diverse characterizations of circle graphs, many of them using the notions of local…
A consistent path system in a graph $G$ is an intersection-closed collection of paths, with exactly one path between any two vertices in $G$. We call $G$ metrizable if every consistent path system in it is the system of geodesic paths…
The book embedding of a graph $G$ is to place the vertices of $G$ on the spine and draw the edges to the pages so that the edges in the same page do not cross with each other. A book embedding is matching if the vertices in the same page…
Due to their flexibility to represent almost any kind of relational data, graph-based models have enjoyed a tremendous success over the past decades. While graphs are inherently only combinatorial objects, however, many prominent analysis…
Let ${\mathcal C}(\Omega)$ be the linear code arising from a projective system $\Omega$ of $\mathrm{PG}(V).$ Consider the point-line geometry $\Gamma=({\mathcal P},{\mathcal L})$ and a projective embedding $\varepsilon\colon…
A multigraph G is triangle decomposable if its edge set can be partitioned into subsets, each of which induces a triangle of G, and rationally triangle decomposable if its triangles can be assigned rational weights such that for each edge e…
A sum graph is a finite simple graph whose vertex set is labeled with distinct positive integers such that two vertices are adjacent if and only if the sum of their labels is itself another label. The spum of a graph $G$ is the minimum…
The basis number of a graph $G$ is the minimum $k$ such that the cycle space of $G$ is generated by a family of cycles using each edge at most $k$ times. A classical result of Mac Lane states that planar graphs are exactly graphs with basis…
The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has a vertex (edge) labeling with $d$ labels that is preserved only by a trivial automorphism. In this paper we consider the maximal…
In the area of beyond-planar graphs, i.e. graphs that can be drawn with some local restrictions on the edge crossings, the recognition problem is prominent next to the density question for the different graph classes. For 1-planar graphs,…