Inspired by the theory of poset games, we introduce a new compound of impartial combinatorial games and provide a complete analysis in the spirit of the Sprague-Grundy theory. Furthermore, we …
Inspired by the theory of poset games, we introduce a new compound of impartial combinatorial games and provide a complete analysis in the spirit of the Sprague-Grundy theory. Furthermore, we establish several substitution and reduction principles for this compound and consider its computational aspects.
A connected undirected graph is called \emph{geodetic} if for every pair of vertices there is a unique shortest path connecting them. It has been conjectured that for finite groups, the …
A connected undirected graph is called \emph{geodetic} if for every pair of vertices there is a unique shortest path connecting them. It has been conjectured that for finite groups, the only geodetic Cayley graphs which occur are odd cycles and complete graphs. In this article we present a series of theoretical results which contribute to a computer search verifying this conjecture for all groups of size up to 1024. The conjecture is also verified theoretically for several infinite families of groups including dihedral and some families of nilpotent groups. Two key results which enable the computer search to reach as far as it does are: if the center of a group has even order, then the conjecture holds (this eliminates all 2-groups from our computer search); if a Cayley graph is geodetic then there are bounds relating the size of the group, generating set and center (which cuts down the number of generating sets which must be searched significantly).
The membership problem for an algebraic structure asks whether a given element is contained in some substructure, which is usually given by generators. In this work we study the membership …
The membership problem for an algebraic structure asks whether a given element is contained in some substructure, which is usually given by generators. In this work we study the membership problem, as well as the conjugacy problem, for finite inverse semigroups. The closely related membership problem for finite semigroups has been shown to be PSPACE-complete in the transformation model by Kozen (1977) and NL-complete in the Cayley table model by Jones, Lien, and Laaser (1976). More recently, both the membership and the conjugacy problem for finite inverse semigroups were shown to be PSPACE-complete in the partial bijection model by Jack (2023). Here we present a more detailed analysis of the complexity of the membership and conjugacy problems parametrized by varieties of finite inverse semigroups. We establish dichotomy theorems for the partial bijection model and for the Cayley table model. In the partial bijection model these problems are in NC (resp. NP for conjugacy) for strict inverse semigroups and PSPACE-complete otherwise. In the Cayley table model we obtain general LOGSPACE-algorithms as well as NPOLYLOGTIME upper bounds for Clifford semigroups and LOGSPACE-completeness otherwise. Furthermore, by applying our findings, we show the following: the intersection non-emptiness problem for inverse automata is PSPACE-complete even for automata with only two states; the subpower membership problem is in NC for every strict inverse semi-group and PSPACE-complete otherwise; the minimum generating set and the equation satisfiability problems are in NP for varieties of finite strict inverse semigroups and PSPACE-complete otherwise.
The membership problem for an algebraic structure asks whether a given element is contained in some substructure, which is usually given by generators. In this work we study the membership …
The membership problem for an algebraic structure asks whether a given element is contained in some substructure, which is usually given by generators. In this work we study the membership problem, as well as the conjugacy problem, for finite inverse semigroups. The closely related membership problem for finite semigroups has been shown to be PSPACE-complete in the transformation model by Kozen (1977) and NL-complete in the Cayley table model by Jones, Lien, and Laaser (1976). More recently, both the membership and the conjugacy problem for finite inverse semigroups were shown to be PSPACE-complete in the partial bijection model by Jack (2023). Here we present a more detailed analysis of the complexity of the membership and conjugacy problems parametrized by varieties of finite inverse semigroups. We establish dichotomy theorems for the partial bijection model and for the Cayley table model. In the partial bijection model these problems are in NC (resp. NP for conjugacy) for strict inverse semigroups and PSPACE-complete otherwise. In the Cayley table model we obtain general LOGSPACE-algorithms as well as NPOLYLOGTIME upper bounds for Clifford semigroups and LOGSPACE-completeness otherwise. Furthermore, by applying our findings, we show the following: the intersection non-emptiness problem for inverse automata is PSPACE-complete even for automata with only two states; the subpower membership problem is in NC for every strict inverse semi-group and PSPACE-complete otherwise; the minimum generating set and the equation satisfiability problems are in NP for varieties of finite strict inverse semigroups and PSPACE-complete otherwise.
A connected undirected graph is called \emph{geodetic} if for every pair of vertices there is a unique shortest path connecting them. It has been conjectured that for finite groups, the …
A connected undirected graph is called \emph{geodetic} if for every pair of vertices there is a unique shortest path connecting them. It has been conjectured that for finite groups, the only geodetic Cayley graphs which occur are odd cycles and complete graphs. In this article we present a series of theoretical results which contribute to a computer search verifying this conjecture for all groups of size up to 1024. The conjecture is also verified theoretically for several infinite families of groups including dihedral and some families of nilpotent groups. Two key results which enable the computer search to reach as far as it does are: if the center of a group has even order, then the conjecture holds (this eliminates all 2-groups from our computer search); if a Cayley graph is geodetic then there are bounds relating the size of the group, generating set and center (which cuts down the number of generating sets which must be searched significantly).
Inspired by the theory of poset games, we introduce a new compound of impartial combinatorial games and provide a complete analysis in the spirit of the Sprague-Grundy theory. Furthermore, we …
Inspired by the theory of poset games, we introduce a new compound of impartial combinatorial games and provide a complete analysis in the spirit of the Sprague-Grundy theory. Furthermore, we establish several substitution and reduction principles for this compound and consider its computational aspects.
Journal Article MOORE GEOMETRIES AND RANK 3 GROUPS HAVING μ=1 Get access WILLIAM M. KANTOR WILLIAM M. KANTOR University of OregonEugene, Oregon 97403 U.S.A. Search for other works by this …
Journal Article MOORE GEOMETRIES AND RANK 3 GROUPS HAVING μ=1 Get access WILLIAM M. KANTOR WILLIAM M. KANTOR University of OregonEugene, Oregon 97403 U.S.A. Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 28, Issue 3, September 1977, Pages 309–328, https://doi.org/10.1093/qmath/28.3.309 Published: 01 September 1977 Article history Received: 13 November 1975 Revision received: 24 September 1976 Published: 01 September 1977
This note treats the existence of connected, undirected graphs homogeneous of degree d and of diameter k, having a number of nodes which is maximal according to a certain definition. …
This note treats the existence of connected, undirected graphs homogeneous of degree d and of diameter k, having a number of nodes which is maximal according to a certain definition. For k = 2 unique graphs exist for d = 2,3,7 and possibly for d = 57 (which is undecided), but for no other degree. For k = 3 a graph exists only for d = 2. The proof exploits the characteristic roots and vectors of the adjacency matrix (and its principal submatrices) of the graph.
Abstract Given a group G and a finite generating set G , we take p G : G → Z to be the function which counts the number of geodesics …
Abstract Given a group G and a finite generating set G , we take p G : G → Z to be the function which counts the number of geodesics for each group element g . This generalizes Pascal's triangle. We compute p G for word hyperbolic and describe generic behaviour in abelian groups.
In this paper, we shall first describe the theory of distance-regular graphs and then apply it to the classification of Moore graphs. The object of the paper is to prove …
In this paper, we shall first describe the theory of distance-regular graphs and then apply it to the classification of Moore graphs. The object of the paper is to prove that there are no Moore graphs (other than polygons) of diameter ≥ 3. An independent proof of this result has been given by Barmai and Ito(1). Taken with the result of (4), this shows that the only possible Moore graphs are the following:
We prove that the groups presented by finite convergent monadic rewriting systems with generators of finite order are exactly the free products of finitely many finite groups, thereby confirming Gilman’s …
We prove that the groups presented by finite convergent monadic rewriting systems with generators of finite order are exactly the free products of finitely many finite groups, thereby confirming Gilman’s conjecture in a special case. We also prove that the finite cyclic groups of order at least three are the only finite groups admitting a presentation by more than one finite convergent monadic rewriting system (up to relabelling), and these admit presentation by exactly two such rewriting systems.
This paper describes a new approach to the problem of generating the class of all geodetic graphs homeomorphic to a given geodetic one. An algorithmic procedure is elaborated to carry …
This paper describes a new approach to the problem of generating the class of all geodetic graphs homeomorphic to a given geodetic one. An algorithmic procedure is elaborated to carry out a systematic finding of such a class of graphs. As a result, the enumeration of the class of geodetic graphs homeomorphic to certain Moore graphs has been performed.
We call a graph k-geodetic, for some [Formula: see text], if it is connected and between any two vertices there are at most k geodesics. It is shown that any …
We call a graph k-geodetic, for some [Formula: see text], if it is connected and between any two vertices there are at most k geodesics. It is shown that any hyperbolic group with a k-geodetic Cayley graph is virtually-free. Furthermore, in such a group the centralizer of any infinite order element is an infinite cyclic group. These results were known previously only in the case that [Formula: see text]. A key tool used to develop the theorem is a new graph theoretic result concerning “ladder-like structures” in a k-geodetic graph.