Related papers: Structural Characterisations of (n-1,n)-Trees
A monotone drawing of a graph G is a straight-line drawing of G such that every pair of vertices is connected by a path that is monotone with respect to some direction. Trees, as a special class of graphs, have been the focus of several…
We analyze some local properties of sparse Erdos-Renyi graphs, where $d(n)/n$ is the edge probability. In particular we study the behavior of very short paths. For $d(n)=n^{o(1)}$ we show that $G(n,d(n)/n)$ has asymptotically almost surely…
Taking a Feynman categorical perspective, several key aspects of the geometry of surfaces are deduced from combinatorial constructions with graphs. This provides a direct route from combinatorics of graphs to string topology operations via…
Geometry of networks endowed with a causal structure is discussed using the conventional framework of equilibrium statistical mechanics. The popular growing network models appear as particular causal models. We focus on a class of tree…
In this paper we give a complete characterization of the statistical equivalence classes of CEGs and of staged trees. We are able to show that all graphical representations of the same model share a common polynomial description. Then,…
Motivated by an approach to visualization of high dimensional statistical data given in Hurley and Oldford (2011), this work examines the clique structure of $J_n(m, m-1)$ Johnson graphs. Cliques and maximal cliques are characterized and…
A construction described by the current author (2017) uses two linear prototypes to build a compound graph with Ramsey properties inherited from the prototype graphs. The resulting graph is linear; and cyclic if both prototypes are cyclic.…
Let ${\cal G}=(G,w)$ be a weighted simple finite connected graph, that is, let $G$ be a simple finite connected graph endowed with a function $w$ from the set of the edges of $G$ to the set of real numbers. For any subgraph $G'$ of $G$, we…
We introduce structured decompositions, category-theoretic structures which simultaneously generalize notions from graph theory (including treewidth, layered treewidth, co-treewidth, graph decomposition width, tree independence number,…
A graph class is monotone if it is closed under taking subgraphs. It is known that a monotone class defined by finitely many obstructions has bounded treewidth if and only if one of the obstructions is a so-called tripod, that is, a…
Generalized trees, we call them O-trees, are defined as hierarchical partial orders, i.e., such that the elements larger than any one are linearly ordered. Quasi-trees are, roughly speaking, undirected O-trees. For O-trees and quasi-trees,…
Let ${\cal T}=(T,w)$ be a weighted finite tree with leaves $1,..., n$.For any $I :=\{i_1,..., i_k \} \subset \{1,...,n\}$, let $D_I ({\cal T})$ be the weight of the minimal subtree of $T$ connecting $i_1,..., i_k$; the $D_{I} ({\cal T})$…
Counting non-isomorphic tree-like multigraphs that include self-loops and multiple edges is an important problem in combinatorial enumeration, with applications in chemical graph theory, polymer science, and network modeling. Traditional…
Transductions are a general formalism for expressing transformations of graphs (and more generally, of relational structures) in logic. We prove that a graph class $\mathscr{C}$ can be $\mathsf{FO}$-transduced from a class of bounded-height…
We give a parity reversing involution on noncrossing trees that leads to a combinatorial interpretation of a formula on noncrossing trees and symmetric ternary trees in answer to a problem proposed by Hough. We use the representation of…
This paper deals with computation trees over an arbitrary structure consisting of a set along with collections of functions and predicates that are defined on it. It is devoted to the comparative analysis of three parameters of problems…
Path graphs are intersection graphs of paths in a tree.~In this paper we give a "6\ good characterization" of path graphs, namely, we prove that path graph membership is in $NP\cap CoNP$ without resorting to existing polynomial time…
We describe the structure of those graphs that have largest spectral radius in the class of all connected graphs with a given degree sequence. We show that in such a graph the degree sequence is non-increasing with respect to an ordering of…
We investigate a structural generalisation of treewidth we call $\mathcal{A}$-blind-treewidth where $\mathcal{A}$ denotes an annotated graph class. This width parameter is defined by evaluating only the size of those bags $B$ of…
A general novel approach mapping discrete, combinatorial, graph-theoretic problems onto ``physical'' models - namely $n$ simplexes in $n-1$ dimensions - is applied to the graph equivalence problem. It is shown to solve this long standing…