We consider graph applications of the well-known paradigm \killing two birds with one stone". In the plane, this gives rise to a stage graph as follows: vertices are the points, and fu; vg is an edge if and only if the (innnite, straight) line segment joining u to v intersects the stage. Such graphs are shown to be comparability graphs of ordered sets of dimension 2. Similar graphs can be const...

Kayles is the game, where two players alternately choose a vertex that has not been chosen before nor is adjacent to an already chosen vertex from a given graph. The last player that choses a vertex wins the game. We show, with help of Sprague-Grundy theory, that the problem to determine which player has a winning strategy for a given graph, can be solved in O(n3) time on interval graphs, on ci...

We investigate the class of reflexive graphs that admit semilattice polymorphisms, and in doing so, give an algebraic characterisation of chordal graphs. In particular, we show that a graph G is chordal if and only if it has a semilattice polymorphism such that G is a subgraph of the comparability graph of the semilattice. Further, we find a new characterisation of the leafage of a chordal grap...

We consider graph applications of the well-known paradigm “killing two birds with one stone”. In the plane, this gives rise to a stage graph as follows: vertices are the points, and {u, II} is an edge if and only if the (infinite, straight) line segment joining u to u intersects the stage. Such graphs are shown to be comparability graphs of ordered sets of dimension 2. Similar graphs can be con...

A sandwich problem for a graph with respect to a graph property Π is a partially specified graph, i.e., only some of the edges and non-edges are given, and the question to be answered is, can this graph be completed to a graph which has the property Π? The graph sandwich problem was investigated for a large number of families of graphs in a 1995 paper by Golumbic, Kaplan and Shamir, and much su...

Philosophical PerspectivesVolume 35, Issue 1 p. 70-98 ORIGINAL ARTICLE Consequences of comparability Cian Dorr, Corresponding Author Dorr [email protected] Email: Jacob M. Nebel [email protected] Jake Zuehl [email protected] for more papers by this authorJacob Nebel, Search authorJake Zuehl, [email protected] author First published: 31 October 2021 https://doi.org/10.1111/phpe.12157Read the full textAbou...

Recent work in mathematical neuroscience has calculated the directed graph homology of simplicial complex given by brains sparse adjacency graph, so called connectome. These biological connectomes show an abundance both high-dimensional simplices and Betti-numbers all viable dimensions - contrast to Erd\H{o}s-R\'enyi-graphs comparable size density. An analysis synthetically trained reveals simi...