Cycling in hypercubes - Marc, Tilen (visokošolska dela, 2018)
We study isometric subgraphs found in hypercubes, called partial cubes. We focus on three aspects: understanding the cycle space of such subgraphs, exploring established subfamilies and properties, and finding symmetric ones. As we show, convex cycles in partial cubes have many intriguing properties, from spanning a simply connected space to forming complex substructures such as intertwinings and traverses. We analyze partial cubes with high girth to obtain results on the structure and degree of such graphs. This knowledge is transferred to symmetric partial cubes to obtain a complete classification of cubic, vertex-transitive ones and to find a connection between partial cubes having mirror automorphisms and finite Coxeter groups. We study various subfamilies of partial cubes to expose a connection between (pseudo-) hyperplane arrangements, antipodal subgraphs, oriented matroids, median graphs, and many other structures found in partial cubes. With our main tool, the concept of partial cube minors, we create a map of partial cubes determining the hierarchical structure of subfamilies of partial cubes, and providing new characterizations and generalizations. Lastly, computational and enumerative properties of partial cubes bounded by their isometric dimension are discussed, together with a result showing that finding isomorphisms of graphs is GI-complete already for one of the simplest classes of partial cubes: median graphs.
Keywords: antipodality; antipodalnost; convex subgraphs; delne kocke; Disertacije; Grafi; graph theory; konveksni podgrafi; matematika; mathematics; metric properties; metrične lastnosti; minorji; minors; oriented matroids; orientirani matroidi; partial cubes; teorija grafov; vertex-transitive graphs; vozliščno tranzitivni grafi