Recognizing graphs of acyclic cubical complexes

AbstractAcyclic 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 O(mlogn) which recognizes whether a given graph G on n vertices with m edges is the graph of an acyclic cubical complex. This is significantly better than the complexity O(mn) of the fastest currently known algorithm for recognizing retracts of cubes in general

Maximal proper subgraphs of median graphs

AbstractFor a median graph G and a vertex v of G that is not a cut-vertex we show that G-v is a median graph precisely when v is not the center of a bipartite wheel, which is in turn equivalent with the existence of a certain edge elimination scheme for edges incident with v. This implies a characterization of vertex-critical (respectively, vertex-complete) median graphs, which are median graphs whose all vertex-deleted subgraphs are not median (respectively, are median). Moreover, two analogous characterizations for edge-deleted median graphs are given

A multifacility location problem on median spaces

AbstractThis paper is concerned with the problem of locating n new facilities in the median space when there are k facilities already located. The objective is to minimize the weighted sum of distances. Necessary and sufficient conditions are established. Based on these results a polynomial algorithm is presented. The algorithm requires the solution of a sequence of minimum-cut problems. The complexity of this algorithm for median graphs and networks and for finite median spaces with ¦V¦points is O(¦V¦3 + ¦V¦ψ(n)), where ψ(n) is the complexity of the applied maximum-flow algorithm. For a simple rectilinear polygon P with N edges and equipped with the rectilinear distance the analogical algorithm requires O(N + k(logN + logk + ψ(n))) time and O(N + kψ(n)) time in the case of the vertex-restricted multifacility location problem

Uprooted Phylogenetic Networks

The need for structures capable of accommodating complex evolutionary signals such as those found in, for example, wheat has fueled research into phylogenetic networks. Such structures generalize the standard model of a phylogenetic tree by also allowing for cycles and have been introduced in rooted and unrooted form. In contrast to phylogenetic trees or their unrooted versions, rooted phylogenetic networks are notoriously difficult to understand. To help alleviate this, recent work on them has also centered on their “uprooted” versions. By focusing on such graphs and the combinatorial concept of a split system which underpins an unrooted phylogenetic network, we show that not only can a so-called (uprooted) 1-nested network N be obtained from the Buneman graph (sometimes also called a median network) associated with the split system  Σ(N)Σ(N)  induced on the set of leaves of N but also that that graph is, in a well-defined sense, optimal. Along the way, we establish the 1-nested analogue of the fundamental “splits equivalence theorem” for phylogenetic trees and characterize maximal circular split systems

Combinatorics and geometry of finite and infinite squaregraphs

Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar dual of a finite squaregraph is determined by a triangle-free chord diagram of the unit disk, which could alternatively be viewed as a triangle-free line arrangement in the hyperbolic plane. This representation carries over to infinite plane graphs with finite vertex degrees in which the balls are finite squaregraphs. Algebraically, finite squaregraphs are median graphs for which the duals are finite circular split systems. Hence squaregraphs are at the crosspoint of two dualities, an algebraic and a geometric one, and thus lend themselves to several combinatorial interpretations and structural characterizations. With these and the 5-colorability theorem for circle graphs at hand, we prove that every squaregraph can be isometrically embedded into the Cartesian product of five trees. This embedding result can also be extended to the infinite case without reference to an embedding in the plane and without any cardinality restriction when formulated for median graphs free of cubes and further finite obstructions. Further, we exhibit a class of squaregraphs that can be embedded into the product of three trees and we characterize those squaregraphs that are embeddable into the product of just two trees. Finally, finite squaregraphs enjoy a number of algorithmic features that do not extend to arbitrary median graphs. For instance, we show that median-generating sets of finite squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure

On embeddings of CAT(0) cube complexes into products of trees

We prove that the contact graph of a 2-dimensional CAT(0) cube complex ${\bf X}$ of maximum degree $\Delta$ can be coloured with at most $\epsilon(\Delta)=M\Delta^{26}$ colours, for a fixed constant $M$. This implies that ${\bf X}$ (and the associated median graph) isometrically embeds in the Cartesian product of at most $\epsilon(\Delta)$ trees, and that the event structure whose domain is ${\bf X}$ admits a nice labeling with $\epsilon(\Delta)$ labels. On the other hand, we present an example of a 5-dimensional CAT(0) cube complex with uniformly bounded degrees of 0-cubes which cannot be embedded into a Cartesian product of a finite number of trees. This answers in the negative a question raised independently by F. Haglund, G. Niblo, M. Sageev, and the first author of this paper.Comment: Some small corrections; main change is a correction of the computation of the bounds in Theorem 1. Some figures repaire

Koreni polinoma kock medianskih grafov

Polinom kock ▫$c(G,X)$▫ grafa ▫$G$▫ je definiran z ▫$sum_{i ge 0}alpha_i(G)x^i$▫, kjer ▫$alpha_i(G)$▫ označuje število induciranih ▫$i$▫-kock v ▫$G$▫. Naj bo ▫$G$▫ medianski graf. Dokazano je, da je vsaka racionalna ničla polinoma ▫$c(G,x)$▫ oblike ▫$-frac{t+1}{t}$▫ za neko celo število ▫$t>0$▫ in da ima ▫$c(G,x)$▫ vedno realno ničlo na intervalu ▫$[-2,-1)$▫. Nadalje ima ▫$c(G,x)$▫ ▫$p$▫-kratno ničlo natanko tedaj, ko je ▫$G$▫ kartezični produkt ▫$p$▫ dreves istega reda. Grafi acikličnih kubičnih kompleksov so karakterizirani kot grafi za katere velja ▫$c(H,-2)=0$▫ za vsak 2-povezan konveksen podgraf ▫$H$▫.The cube polynomial ▫$c(G,X)$▫ of a graph ▫$G$▫ is defined as ▫$sum_{i ge 0}alpha_i(G)x^i$▫, where ▫$alpha_i(G)$▫ denotes the number of induced ▫$i$▫-cubes of ▫$G$▫, in particular, ▫$alpha_0(G) = |V(G)|$▫ and ▫$alpha_1(G) = |E(G)|$▫. Let ▫$G$▫ be a median graph. It is proved that every rational zero of ▫$c(G,x)$▫ is of the form ▫$-frac{t+1}{t}$▫ for some integer ▫$t>0$▫ and that ▫$c(G,x)$▫ always has a real zero in the interval ▫$[-2,-1)$▫. Moreover, ▫$c(G,x)$▫ has a ▫$p$▫-multiple zero if and only if ▫$G$▫ is the cartesian product of ▫$p$▫ trees all of the same order. Graphs of acyclic cubical complexes are characterized as the graphs ▫$G$▫ for which ▫$c(H,-2)=0$▫ holds for every 2-connected convex subgraph ▫$H$▫ of ▫$G$▫. Median graphs that are Cartesian products are also characterized