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A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines $l_1$ and $l_2$, one on each. A bipartite permutation graph is a permutation graph which is bipartite. In this paper we study…
This work addresses the intrinsic relationship between trees and networks (i.e. graphs). A complete (invertible) mapping is presented which allows trees to be mapped into weighted graphs and then backmapped into the original tree without…
The simple connected graphs may be classified by their cycle composition (number and lengths of cycles). This work derives the counting series of the simple connected graphs that have cycles of unrestricted number and length, but no…
A \textit{distinguishing partition} of a group $X$ with automorphism group ${aut}(X)$ is a partition of $X$ that is fixed by no nontrivial element of ${aut}(X)$. In the event that $X$ is a complete multipartite graph with its automorphism…
A rooted tree is balanced if the degree of a vertex depends only on its distance to the root. In this paper we determine the sharp threshold for the appearance of a large family of balanced spanning trees in the random geometric graph…
Gallai-colorings are edge-colored complete graphs in which there are no rainbow triangles. Within such colored complete graphs, we consider Ramsey-type questions, looking for specified monochromatic graphs. In this work, we consider…
Let $T(n,m)$ be the set of all plane labelled bipartite trees with $n$ white vertices and $m$ black. If the number $n+m$ of vertices is even, then the set $T(n,m)$ is a union of two disjoint subsets --- subset od "even" trees and subset of…
We generalize some homotopy calculation techniques such as splittings and matching trees that are introduced for the computations in the case of the independence complexes of graphs to arbitrary simplicial complexes, and exemplify their…
A spin system is a framework in which the vertices of a graph are assigned spins from a finite set. The interactions between neighbouring spins give rise to weights, so a spin assignment can also be viewed as a weighted graph homomorphism.…
Intuitively speaking, a bipartite graph is mirror if it can be drawn in the Cartesian plane in such a way that, the vertices of one stable are points in x=0, the vertices of the other stable set are points in x=1, the edges are straight…
The treewidth of a graph is an important invariant in structural and algorithmic graph theory. This paper studies the treewidth of line graphs. We show that determining the treewidth of the line graph of a graph $G$ is equivalent to…
A graph is near-bipartite if its vertex set can be partitioned into an independent set and a set which induces a forest. In this paper, planar graphs without cycles of length from 4 to 7 are shown to be near-bipartite.
Extending the work of Godsil and others, we investigate the notion of the inverse of a graph (specifically, of bipartite graphs with a unique perfect matching). We provide a concise necessary and sufficient condition for the invertibility…
Linear arrangements of graphs are a well-known type of graph labeling and are found in many important computational problems, such as the Minimum Linear Arrangement Problem ($\texttt{minLA}$). A linear arrangement is usually defined as a…
A fullerene graph is a cubic bridgeless planar graph with twelve 5-faces such that all other faces are 6-faces. We show that any fullerene graph on n vertices can be bipartized by removing O(sqrt{n}) edges. This bound is asymptotically…
A graph $G$ is said to be a `set graph' if it admits an acyclic orientation that is also `extensional', in the sense that the out-neighborhoods of its vertices are pairwise distinct. Equivalently, a set graph is the underlying graph of the…
A word-representable graph is a simple graph $G$ which can be represented by a word $w$ over the vertices of $G$ such that any two vertices are adjacent in $G$ if and only if they alternate in $w$. It is known that the class of…
A split graph is a graph whose vertices can be partitioned into a clique and a stable set. We investigate the combinatorial species of split graphs, providing species-theoretic generalizations of enumerative results due to B\'ina and…
We define the \emph{visual complexity} of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to…
We explore the concept of separating systems of vertex sets of graphs. A separating system of a set $X$ is a collection of subsets of $X$ such that for any pair of distinct elements in $X$, there exists a set in the separating system that…