Related papers: Path abstraction
A graph is \emph{hamiltonian-connected} if every pair of vertices can be connected by a hamiltonian path, and it is \emph{hamiltonian} if it contains a hamiltonian cycle. We construct families of non-hamiltonian graphs for which the ratio…
The famous Gallai's Conjecture states that any connected graph with n vertices has a path decomposition containing at most (n+1)/2 paths. In this note, we explore graphs generated from removing edges from complete graphs. We first provide…
What does a typical road network look like? Existing generative models tend to focus on one aspect to the exclusion of others. We introduce the general-purpose \emph{quadtree model} and analyze its shortest paths and maximum flow.
We introduce a new combinatorial abstraction for the graphs of polyhedra. The new abstraction is a flexible framework defined by combinatorial properties, with each collection of properties taken providing a variant for studying the…
A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that "any graph is close to being the disjoint union of expanders". Our goal in this paper…
Classical path search assumes complete graphs and scalar optimization metrics, yet real infrastructure networks are incomplete and require multi-dimensional evaluation. We introduce the concept of traversal: a generalization of paths that…
Stratified digraphs are popular models for feedforward neural networks. However, computation of their path homologies has been limited to low dimensions due to high computational complexity. A recursive algorithm is proposed to compute…
A broader definition of generalized truncations of graphs is introduced followed by an exploration of some standard concepts and parameters with regard to generalized truncations.
In this paper, we consider the problem of counting and sampling structures in graphs. We define a class of "edge universal labeling problems"---which include proper $k$-colorings, independent sets, and downsets---and describe simple…
We show that all the tangles in a finite graph or matroid can be distinguished by a single tree-decomposition that is invariant under the automorphisms of the graph or matroid. This comes as a corollary of a similar decomposition theorem…
We organize a table of regular graphs with minimal diameters and minimal mean path lengths, large bisection widths and high degrees of symmetries, obtained by enumerations on supercomputers. These optimal graphs, many of which are newly…
This introduction to graphs and graph algebras provides the optimal bound for the number of all paths of length $k$ in a graph with $N\geq k$ edges and no loops. Our proof relies on a construction of a number of terminating algorithms that…
The transmission of a connected hypergraph is defined as the summation of distances between all unordered pairs of distinct vertices. We determine the unique uniform unicyclic hypergraphs of fixed size with minimum and maximum…
We prove that in any $n$-vertex complete graph there is a collection $\mathcal{P}$ of $(1 + o(1))n$ paths that strongly separates any pair of distinct edges $e, f$, meaning that there is a path in $\mathcal{P}$ which contains $e$ but not…
We study a type of object, called a pathway (generalizing pathways in the sense of P. E. Cohen [Proc. Amer. Math. Soc. 74, No. 2 (1979), 318--321]), which is useful for several set-theoretic constructions and whose existence, in a sense,…
Let $E \subset {\Bbb F}_q^d$, the $d$-dimensional vector space over a finite field with $q$ elements. Construct a graph, called the distance graph of $E$, by letting the vertices be the elements of $E$ and connect a pair of vertices…
We investigate connections between the symmetries (automorphisms) of a graph and its spectral properties. Whenever a graph has a symmetry, i.e. a nontrivial automorphism $\phi$, it is possible to use $\phi$ to decompose any matrix…
A graph drawing in the plane is called an almost embedding if the images of any two non-adjacent simplices (i.e. vertices or edges) are disjoint. Almost embeddings (more precisely, their higher-dimensional analogues) naturally appear in…
A traversal of a connected graph is a linear ordering of its vertices all of whose initial segments induce connected subgraphs. Traversals, and their refinements such as breadth-first and depth-first traversals, are computed by various…
A {\bf map} is a graph that admits an orientation of its edges so that each vertex has out-degree exactly 1. We characterize graphs which admit a decomposition into $k$ edge-disjoint maps after: (1) the addition of {\it any} $\ell$ edges;…