arXiv subjects

Dana C. Ernst

Publications and source records attributed to Dana C. Ernst.

At least 19 recordsLinked to original sources

arXiv2026

Impartial removing games on grid graphs

A subset of the vertex set of a graph is geodetically convex if it contains every vertex on any shortest path between two elements of the subset. The convex hull of a set of vertices is the smallest convex set containing the set. We study two games in which two players take turns selecting vertices of a graph. The last player to move is the winner. The achievement game ends when the convex hull of the unselected vertices does not contain every vertex in the graph. In the avoidance game, the convex hull of the remaining vertices must contain every vertex. We determine the nim-value of these games for the family of grid graphs. We also provide some results for lattice graphs. Key tools in this analysis are delayed gamegraphs, option preserving maps, and case analysis diagrams.

math.CO↗

arXiv2026

Impartial geodetic removing games on graphs

A subset of the vertex set of a graph is geodetically convex if it contains every vertex on any shortest path between two elements of the subset. The convex hull of a set of vertices is the smallest convex set containing the set. We study two games in which two players take turns selecting vertices of a graph until the convex hull of the jointly unselected vertices is too small. The last player to move is the winner. The achievement game ends when the convex hull of the jointly unselected vertices is not the vertex set. In the avoidance game, the convex hull of the jointly unselected vertices must always be the vertex set. We study the nim-values for several graph families, including cycle graphs, hypercube graphs, complete multipartite graphs, wheel graphs, generalized wheel graphs, and graphs with a unique minimal generating set.

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arXiv2025

Isomorphism Theorems for Impartial Combinatorial Games

We introduce the category of optiongraphs and option-preserving maps as a model to study impartial combinatorial games. Outcomes, remoteness, and extended nim-values are preserved under option-preserving maps. We show that the four isomorphism theorems from universal algebra are valid in this category. Quotient optiongraphs, including the minimum quotient, provide simplifications that can help in the analysis of games.

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arXiv2024

Categories of impartial rulegraphs and gamegraphs

The traditional mathematical model for an impartial combinatorial game is defined recursively as a set of the options of the game, where the options are games themselves. We propose a model called gamegraph, together with its generalization rulegraph, based on the natural description of a game as a digraph where the vertices are positions and the arrows represent possible moves. Such digraphs form a category where the morphisms are option preserving maps. We study several versions of this category. Our development includes congruence relations, quotients, and isomorphism theorems and is analogous to the corresponding notions in universal algebra. The quotient by the maximum congruence relation produces an object that is essentially equivalent to the traditional model. After the development of the general theory, we count the number of non-isomorphic gamegraphs and rulegraphs by formal birthday and the number of positions.

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arXiv2024

Impartial geodetic building games on graphs

A subset of the vertex set of a graph is geodetically convex if it contains every vertex on any shortest path between two elements of the set. The convex hull of a set of vertices is the smallest convex set containing the set. We study variations of two games introduced by Buckley and Harary, where two players take turns selecting previously-unselected vertices of a graph until the convex hull of the jointly-selected vertices becomes too large. The last player to move is the winner. The achievement game ends when the convex hull contains every vertex. In the avoidance game, the convex hull is not allowed to contain every vertex. We determine the nim-value of these games for several graph families.

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arXiv2024

Braid graphs in simply-laced triangle-free Coxeter systems are partial cubes

In this paper, we study the structure of braid graphs in simply-laced Coxeter systems. We prove that every reduced expression has a unique factorization as a product of so-called links, which in turn induces a decomposition of the braid graph into a box product of the braid graphs for each link factor. When the Coxeter graph has no three-cycles, we use the decomposition to prove that braid graphs are partial cubes, i.e., can be isometrically embedded into a hypercube. For a special class of links, called Fibonacci links, we prove that the corresponding braid graphs are Fibonacci cubes.

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arXiv2024

Braid graphs in simply-laced triangle-free Coxeter systems are median

Any two reduced expressions for the same Coxeter group element are related by a sequence of commutation and braid moves. Two reduced expressions are said to be braid equivalent if they are related via a sequence of braid moves. Braid equivalence is an equivalence relation and the corresponding equivalence classes are called braid classes. Each braid class can be encoded in terms of a braid graph in a natural way. In a recent paper, Awik et al.~proved that when a Coxeter system is simply laced and triangle free (i.e., the corresponding Coxeter graph has no three-cycles), the braid graph for a reduced expression is a partial cube (i.e., isometric to a subgraph of a hypercube). In this paper, we will provide an alternate proof of this fact, as well as determine the minimal dimension hypercube into which a braid graph can be isometrically embedded, which addresses an open question posed by Awik et al. For our main result, we prove that braid graphs in simply-laced triangle-free Coxeter systems are median, which is a strengthening of previous results.

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arXiv2024

Pattern-avoiding Cayley permutations via combinatorial species

A Cayley permutation is a word of positive integers such that if a letter appears in this word, then all positive integers smaller than that letter also appear. We initiate a systematic study of pattern avoidance on Cayley permutations adopting a combinatorial species approach. Our methods lead to species equations, generating series, and counting formulas for Cayley permutations avoiding any pattern of length at most three. We also introduce the species of primitive structures as a generalization of Cayley permutations with no "flat steps". Finally, we explore various notions of Wilf equivalence arising in this context.

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arXiv2023

The spectrum of nim-values for achievement games for generating finite groups

We study an impartial achievement game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The game ends when the jointly selected elements generate the group. The last player able to make a move is the winner of the game. We prove that the spectrum of nim-values of these games is $\{0,1,2,3,4\}$. This positively answers two conjectures from a previous paper by the last two authors.

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arXiv2018

Impartial achievement games for generating nilpotent groups

We study an impartial game introduced by Anderson and Harary. The game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins. We determine the nim-numbers of this game for finite groups of the form $T \times H$, where $T$ is a $2$-group and $H$ is a group of odd order. This includes all nilpotent and hence abelian groups.

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arXiv2018

Non-cancellable elements in type affine $C$ Coxeter groups

Let $(W,S)$ be a Coxeter system and suppose that $w \in W$ is fully commutative (in the sense of Stembridge) and has a reduced expression beginning (respectively, ending) with $s \in S$. If there exists $t\in S$ such that $s$ and $t$ do not commute and $tw$ (respectively, $wt$) is no longer fully commutative, we say that $w$ is left (respectively, right) weak star reducible by $s$ with respect to $t$. In this paper, we classify the fully commutative elements in Coxeter groups of types $B$ and affine $C$ that are irreducible under weak star reductions. In a sequel to this paper, the classification of the weak star irreducible elements in a Coxeter system of type affine $C$ will provide the groundwork for inductive arguments used to prove the faithfulness of a generalized Temperley--Lieb algebra of type affine $C$ by a particular diagram algebra.

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arXiv2017

Diagram calculus for a type affine $C$ Temperley--Lieb algebra, II

In a previous paper, we presented an infinite dimensional associative diagram algebra that satisfies the relations of the generalized Temperley--Lieb algebra having a basis indexed by the fully commutative elements of the Coxeter group of type affine $C$. We also provided an explicit description of a basis for the diagram algebra. In this paper, we show that the diagrammatic representation is faithful and establish a correspondence between the basis diagrams and the so-called monomial basis of the Temperley--Lieb algebra of type affine $C$.

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arXiv2017

Impartial achievement and avoidance games for generating finite groups

We study two impartial games introduced by Anderson and Harary and further developed by Barnes. Both games are played by two players who alternately select previously unselected elements of a finite group. The first player who builds a generating set from the jointly selected elements wins the first game. The first player who cannot select an element without building a generating set loses the second game. After the development of some general results, we determine the nim-numbers of these games for abelian and dihedral groups. We also present some conjectures based on computer calculations. Our main computational and theoretical tool is the structure diagram of a game, which is a type of identification digraph of the game digraph that is compatible with the nim-numbers of the positions. Structure diagrams also provide simple yet intuitive visualizations of these games that capture the complexity of the positions.

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arXiv2017

Impartial achievement games for generating generalized dihedral groups

We study an impartial game introduced by Anderson and Harary. This game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins. We determine the nim-numbers of this game for generalized dihedral groups, which are of the form $\operatorname{Dih}(A)= \mathbb{Z}_2 \ltimes A$ for a finite abelian group $A$.

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arXiv2016

Factorization of Temperley--Lieb diagrams

The Temperley--Lieb algebra is a finite dimensional associative algebra that arose in the context of statistical mechanics and occurs naturally as a quotient of the Hecke algebra arising from a Coxeter group of type $A$. It is often realized in terms of a certain diagram algebra, where every diagram can be written as a product of "simple diagrams." These factorizations correspond precisely to factorizations of the so-called fully commutative elements of the Coxeter group that index a particular basis. Given a reduced factorization of a fully commutative element, it is straightforward to construct the corresponding diagram. On the other hand, it is generally difficult to reconstruct the factorization given an arbitrary diagram. We present an efficient algorithm for obtaining a reduced factorization for a given diagram.

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arXiv2016

Impartial avoidance and achievement games for generating symmetric and alternating groups

We study two impartial games introduced by Anderson and Harary. Both games are played by two players who alternately select previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins the first game. The first player who cannot select an element without building a generating set loses the second game. We determine the nim-numbers, and therefore the outcomes, of these games for symmetric and alternating groups.

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arXiv2016

On the number of commutation classes of the longest element in the symmetric group

Using the standard Coxeter presentation for the symmetric group $S_n$, two reduced expressions for the same group element are said to be commutation equivalent if we can obtain one expression from the other by applying a finite sequence of commutations. The resulting equivalence classes of reduced expressions are called commutation classes. How many commutation classes are there for the longest element in $S_n$?

math.HO↗

arXiv2016

Impartial avoidance games for generating finite groups

We study an impartial avoidance game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The first player who cannot select an element without making the set of jointly-selected elements into a generating set for the group loses the game. We develop criteria on the maximal subgroups that determine the nim-numbers of these games and use our criteria to study our game for several families of groups, including nilpotent, sporadic, and symmetric groups.

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