On the complexity of testing a graph for n-cube

In the context of reconstructing phylogenetic networks from a collection of phylogenetic trees, several characterisations and subsequently algorithms have been established to reconstruct a phylogenetic network that collectively embeds all trees in the input in some minimum way. For many instances however, the resulting network also embeds additional phylogenetic trees that are not part of the input. However, little is known about these inferred trees. In this paper, we explore the relationships among all phylogenetic trees that are embedded in a given phylogenetic network. First, we investigate some combinatorial properties of the collection $\mathcal{P}$ of all rooted binary phylogenetic trees that are embedded in a rooted binary phylogenetic network $\mathcal{N}$. To this end, we associated a particular graph G, which we call rSPR graph, with the elements in $\mathcal{P}$ and show that, if $|\mathcal{P}|=2^k$, where k is the number of vertices with in-degree two in $\mathcal{N}$, then G has a Hamilton cycle. Second, by exploiting rSPR graphs and properties of hypercubes, we turn to the well-studied class of rooted binary level-1 networks and give necessary and sufficient conditions for when a set of rooted binary phylogenetic trees can be embedded in a level-1 network without inferring any additional trees. Lastly, we show how these conditions translate into a polynomial-time algorithm to reconstruct such a network if it exists.

Acyclic cubical complexes have first been introduced by Bandelt and Chepoi in analogy to acyclic simplicial complexes. They characterized them by cube contraction and elimination schemes and showed that the graphs of acyclic cubical complexes are retracts of cubes characterized by certain forbidden convex subgraphs. In this paper we present an algorithm of time complexity $\text{O}\text{(m}\text{log}\text{n)}$ which recognizes whether a given graph $\text{G}$ on $\text{n}$ vertices with $\text{m}$ edges is the graph of an acyclic cubical complex. This is significantly better than the complexity $\text{O}\text{(m}\text{n}\text{)}$ of the fastest currently known algorithm for recognizing retracts of cubes in general.

A hierarchy of classes of graphs is proposed which includes hypercubes, acyclic cubical complexes, median graphs, almost-median graphs, semi-median graphs and partial cubes. Structural properties of these classes are derived and used for the characterization of these classes by expansion procedures, for a characterization of semi-median graphs by metrically defined relations on the edge set of a graph and for a characterization of median graphs by forbidden subgraphs. Moreover, a convexity lemma is proved and used to derive a simple algorithm of complexityO(mn) for recognizing median graphs.

Hamming graphs are, by definition, the Cartesian product of complete graphs. In the bipartite case these graphs are hypercubes. We present an algorithm recognizing Hamming graphs in linear time and space. This improves a previous algorithm which was linear in time but not in space. This also favorably compares to the general decomposition algorithms of graphs with respect to the Cartesian product, none of which is linear.

This paper contains a new algorithm that recognizes whether a given graphGis a Hamming graph, i.e. a Cartesian product of complete graphs, inO(m)time andO(n²)space. Heremandndenote the numbers of edges and vertices ofG,respectively. Previously this was only possible inO(mlogn)time.

Moreover, we present a survey of other recognition algorithms for Hamming graphs, retracts of Hamming graphs and isometric subgraphs of Hamming graphs. Special emphasis is also given to the bipartite case in which these classes are reduced to binary Hamming graphs, median graphs and partial binary Hamming graphs.

Graphs that can be isometrically embedded into the metric space l₁ are called l₁-graphs. Halved cubes play an important role in the characterization of l₁-graphs. We present an algorithm that recognizes halved cubes in O(n log² n) time.