Masahiro Hachimori

A comment on pure-strategy Nash equilibria in competitive diffusion games

In [N. Alon, M. Feldman, A.D. Procaccia, M. Tennenholtz, A note on competitive diffusion through social networks, Inform. Process. Lett. 110 (2010) 221-225], the authors introduced a game-theoretic model of diffusion process through a network. They showed a relation between the diameter of a given network and existence of pure Nash equilibria in the game. Theorem 1 of their paper says that a pure Nash equilibrium exists if the diameter is at most two. However, we have an example which does not admit a pure Nash equilibrium even if the diameter is two. Hence we correct the statement of Theorem 1 of their paper.

Nonconstructible Simplicial Balls and a Way of Testing Constructibility

Abstract. Constructibility of simplicial complexes is a notion weaker than shellability. It is known that shellable pseudomanifolds are homeomorphic to balls or spheres but simplicial complexes homeomorphic to balls or spheres need not be shellable in general. Constructible pseudomanifolds are also homeomorphic to balls or spheres, but the existence of nonconstructible balls was not known. In this paper we study the constructibility of some nonshellable balls and show that some of them are not constructible, either. Moreover, we give a necessary and sufficient condition for the constructibility of three-dimensional simplicial balls, whose vertices are all on the boundary.

Note: Decompositions of two-dimensional simplicial complexes

We show that the class of Cohen-Macaulay complexes, that of complexes with constructible subdivisions, and that of complexes with shellable subdivisions differ from each other in every dimension d>=2. Further, we give a characterization of two-dimensional simplicial complexes with shellable subdivisions, and show also that they are constructible.

Non-constructible Complexes and the Bridge Index

We show that if a three-dimensional polytopal complex has a knot in its 1-skeleton, where the bridge index of the knot is larger than the number of edges of the knot, then the complex is not constructible, and hence, not shellable. As an application we settle a conjecture of Hetyei concerning the shellability of cubical barycentric subdivisions of 3-spheres. We also obtain similar bounds concluding that a 3-sphere or 3-ball is non-shellable or not vertex decomposable. These two last bounds are sharp.

Compact Grid Representation of Graphs

A graph G is said to be grid locatable if it admits a representation such that vertices are mapped to grid points and edges to line segments that avoid grid points but the extremes. Additionally G is said to be properly embeddable in the grid if it is grid locatable and the segments representing edges do not cross each other. We study the area needed to obtain those representations for some graph families.

Obstructions to shellability, partitionability, and sequential Cohen-Macaulayness

For a property P of simplicial complexes, a simplicial complex @C is an obstruction to P if @C itself does not satisfy P but all of its proper restrictions satisfy P. In this paper, we determine all obstructions to shellability of dimension =<2, refining the previous work by Wachs. As a consequence we obtain that the set of obstructions to shellability, that to partitionability and that to sequential Cohen-Macaulayness all coincide for dimensions =<2. We also show that these three sets of obstructions coincide in the class of flag complexes. These results show that the three properties, hereditary-shellability, hereditary-partitionability, and hereditary-sequential Cohen-Macaulayness are equivalent for these classes.

h-Assignments of simplicial complexes and reverse search

There is currently no efficient algorithm for deciding whether a given simplicial complex is shellable. We propose a practical method that decides shellability of simplicial complexes based on reverse search, which improves an earlier attempt by Moriyama, Nagai and Imai. We also propose to use Macaulays theorem during the search. This works efficiently in high-dimensional cases.

Deciding constructibility of 3—balls with at most two interior vertices

Abstract In this paper, we treat the problem to find an efficient algorithm to decide constructibility. Such an algorithm was given only under the condition that the given simplicial complex is a triangulated 3-ball with all the vertices on the boundary [M. Hachimori, Non-constructible simplicial balls and a way of testing constructibility, Discrete Comput. Geom. 22 (1999) 223–230]. Here we extend this result to the case that the triangulated 3-ball has at most two interior vertices. Our algorithm runs in O(#facets) time. Also, we give an example which shows that the same strategy cannot be used for the cases with more than two interior vertices.

A note on shellability and acyclic orientations

In this short note we discuss the shellability of (nonpure) simplicial complexes in terms of acyclic orientations of the facet-ridge incidence graphs, which shows that we can decide shellability only from the facet-ridge incidences and the total number of faces the simplicial complex contains.

Note: On the topology of the free complexes of convex geometries

We investigate the free complex of a convex geometry. Edelman and Reiner showed that the free complex of a convex geometry is contractible. Moreover, their paper stated a conjecture about the topology of free complexes. This paper proves their conjecture.