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N-matrices are real $n\times n$ matrices all of whose principal minors are negative. We provide (i) an $O(2^n)$ test to detect whether or not a given matrix is an N-matrix, and (ii) a characterization of N-matrices, leading to the recursive…
To determine that two given undirected graphs are isomorphic, we construct for them auxiliary graphs, using the breadth-first search. This makes capability to position vertices in each digraph with respect to each other. If the given graphs…
The fractional perfect b-matching polytope of an undirected graph G is the polytope of all assignments of nonnegative real numbers to the edges of G such that the sum of the numbers over all edges incident to any vertex v is a prescribed…
Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its $1$-skeleton. Call a vertex of a $d$-polytope \emph{nonsimple} if the number of edges incident to it is more than $d$.…
Our input is a complete graph $G = (V,E)$ on $n$ vertices where each vertex has a strict ranking of all other vertices in $G$. Our goal is to construct a matching in $G$ that is popular. A matching $M$ is popular if $M$ does not lose a…
An n-gon is defined as a sequence \P=(V_0,...,V_{n-1}) of n points on the plane. An n-gon \P is said to be convex if the boundary of the convex hull of the set {V_0,...,V_{n-1}} of the vertices of \P coincides with the union of the edges…
We investigate when a complete graph $K_n$ with some edges deleted is determined by its adjacency spectrum. It is shown to be the case if the deleted edges form a matching, a complete graph $K_m$ provided $m \leq n-2$, or a complete…
We investigate some combinatorial properties of convex polytopes simple in edges. For polytopes whose nonsimple vertices are located sufficiently far one from another, we prove an analog of the Hard Lefschetz theorem. It implies Stanley's…
A connected graph $G$ with at least two vertices is matching covered if each of its edges lies in a perfect matching. A matching covered graph is minimal if the removal of any edge results in a graph that is no longer matching covered. An…
Generically, the set of points along which two non-singular vector fields on the three-sphere are positively (resp. negatively) collinear form a link. We prove that the two vector fields are homotopic if and only if the linking number of…
We study the complexity of proving that a sparse random regular graph on an odd number of vertices does not have a perfect matching, and related problems involving each vertex being matched some pre-specified number of times. We show that…
Let $a,b$ be elements in a unital C$^*$-algebra with $0\leq a,b\leq 1$. The element $a$ is absolutely compatible with $b$ if $$\vert a - b \vert + \vert 1 - a - b \vert = 1.$$ In this note we find some technical characterizations of…
It is proved that any polynomial vector field in two complex variables which is complete on a non-algebraic trajectory is complete.
Hladky, Hu, and Piguet [Tilings in graphons, preprint] introduced the notions of matching and fractional vertex covers in graphons. These are counterparts to the corresponding notions in finite graphs. Combinatorial optimization studies the…
Bipartite testing has been a central problem in the area of property testing since its inception in the seminal work of Goldreich, Goldwasser and Ron [FOCS'96 and JACM'98]. Though the non-tolerant version of bipartite testing has been…
A polyhedral norm is a norm N on R^n for which the set N(x)\leq 1 is a polytope. This covers the case of the L^1 and L^{\infty} norms. We consider here effective algorithms for determining the Voronoi polytope for such norms with a point…
Polypols are natural generalizations of polytopes, with boundaries given by nonlinear algebraic hypersurfaces. We describe polypols in the plane and in 3-space that admit a unique adjoint hypersurface and study them from an…
We describe a provably complete algorithm for the generation of a tight, possibly exact superset of all combinatorially distinct simple n-facet polytopes in R^d, along with their graphs, f-vectors, and face lattices. The technique applies…
A popular method in combinatorial optimization is to express polytopes P, which may potentially have exponentially many facets, as solutions of linear programs that use few extra variables to reduce the number of constraints down to a…
We consider the method of alternating projections for finding a point in the intersection of two closed sets, possibly nonconvex. Assuming only the standard transversality condition (or a weaker version thereof), we prove local linear…