Related papers: Vertex decomposability of complexes associated to …
We show that every connected graph can be approximated by a normal tree, up to some arbitrarily small error phrased in terms of neighbourhoods around its ends. The existence of such approximate normal trees has consequences of both…
Consider a group G and a family $\mathcal{A}$ of subgroups of G. We say that vertex finiteness holds for splittings of G over $\mathcal{A}$ if, up to isomorphism, there are only finitely many possibilities for vertex stabilizers of minimal…
The theory of complexes of directed trees was initiated by Kozlov to answer a question by Stanley, and later on, results from the theory were used by Babson and Kozlov in their proof of the Lovasz conjecture. We develop the theory and prove…
A dense forest is a set $F \subset \mathbb{R}^n$ with the property that for all $\varepsilon > 0$ there exists a number $V(\varepsilon) > 0$ such that all line segments of length $V(\varepsilon)$ are $\varepsilon$-close to a point in $F$.…
A subset $M$ of a continuum $X$ is called a \textit{meager composant} if $M$ is maximal with respect to the property that every two of its points are contained in a nowhere dense subcontinuum of $X$. Motivated by questions of Bellamy,…
We use convex decomposition theory to (1) reprove the existence of a universally tight contact structure on every irreducible 3-manifold with nonempty boundary, and (2) prove that every toroidal 3-manifold carries infinitely many…
Working with tree graphs is always easier than with loopy ones and spanning trees are the closest tree-like structures to a given graph. We find a correspondence between the solutions of random K-satisfiability problem and those of spanning…
We study minimal vertex covers of trees. Contrarily to the number $N_{vc}(A)$ of minimal vertex covers of the tree $A$, $\log N_{vc}(A)$ is a self-averaging quantity. We show that, for large sizes $n$, $\lim_{n\to +\infty} <\log…
We develop tools for characterizing vertices of fiber products of polytopes and apply them to simplicial distribution polytopes, a class of probability polytopes arising in quantum foundations and quantum information. In the theory of…
In this note, we classify all the weighted oriented forests whose edge ideals have the property that one of their matching powers has linear resolution.
We characterize the ``best'' model geometries for the class of virtually free groups, and we show that there is a countable infinity of distinct ``best'' model geometries in an appropriate sense--these are the maximally symmetric trees. The…
We consider combinatorial problems that can be solved in polynomial time for graphs of bounded treewidth but where the order of the polynomial that bounds the running time is expected to depend on the treewidth bound. First we review some…
We prove that the class of trees with unique minimum edge-vertex dominating sets is equivalent to the class of trees with unique minimum paired dominating sets.
We consider special cases of the two tree degree sequences problem. We show that if two tree degree sequences do not have common leaves then they always have edge-disjoint caterpillar realizations. By using a probabilistic method, we prove…
Patches of vegetation consist of dense clusters of shrubs, grass, or trees, often found to be circular characteristic size, defined by the properties of the vegetation and terrain. Therefore, vegetation patches can be interpreted as…
Considering the wired uniform spanning forest on a nonunimodular transitive graph, we show that almost surely each tree of the wired uniform spanning forest is light. More generally we study the tilted volumes for the trees in the wired…
Matchings and coverings are central topics in graph theory. The close relationship between these two has been key to many fundamental algorithmic and polyhedral results. For mixed graphs, the notion of matching forest was proposed as a…
We prove that a continuum $X$ is tree-like (resp. circle-like, chainable) if and only if for each open cover $\U_4=\{U_1,U_2,U_3,U_4\}$ of $X$ there is a $\U_4$-map $f:X\to Y$ onto a tree (resp. onto the circle, onto the interval). A…
We show that several new classes of groups are measure strongly treeable. In particular, finitely generated groups admitting planar Cayley graphs, elementarily free groups, and the group of isometries of the hyperbolic plane and all its…
We reprove a result of Dehkordi, Frati, and Gudmundsson: every two vertices in a non-obtuse triangulation of a point set are connected by an angle-monotone path--an xy-monotone path in an appropriately rotated coordinate system. We show…