Strongly simplicial vertices of powers of trees

AbstractFor a tree T and an integer k⩾1, it is well known that the kth power Tk of T is strongly chordal and hence has a strong elimination ordering of its vertices. In this note we obtain a complete characterization of strongly simplicial vertices of Tk, thereby characterizing all strong elimination orderings of the vertices of Tk

On strongly chordal graphs that are not leaf powers

A common task in phylogenetics is to find an evolutionary tree representing proximity relationships between species. This motivates the notion of leaf powers: a graph G = (V, E) is a leaf power if there exist a tree T on leafset V and a threshold k such that uv is an edge if and only if the distance between u and v in T is at most k. Characterizing leaf powers is a challenging open problem, along with determining the complexity of their recognition. This is in part due to the fact that few graphs are known to not be leaf powers, as such graphs are difficult to construct. Recently, Nevries and Rosenke asked if leaf powers could be characterized by strong chordality and a finite set of forbidden subgraphs. In this paper, we provide a negative answer to this question, by exhibiting an infinite family \G of (minimal) strongly chordal graphs that are not leaf powers. During the process, we establish a connection between leaf powers, alternating cycles and quartet compatibility. We also show that deciding if a chordal graph is \G-free is NP-complete, which may provide insight on the complexity of the leaf power recognition problem

Spectral rigidity of automorphic orbits in free groups

It is well-known that a point $T\in cv_N$ in the (unprojectivized) Culler-Vogtmann Outer space $cv_N$ is uniquely determined by its \emph{translation length function} $||.||_T:F_N\to\mathbb R$. A subset $S$ of a free group $F_N$ is called \emph{spectrally rigid} if, whenever $T,T'\in cv_N$ are such that $||g||_T=||g||_{T'}$ for every $g\in S$ then $T=T'$ in $cv_N$. By contrast to the similar questions for the Teichm\"uller space, it is known that for $N\ge 2$ there does not exist a finite spectrally rigid subset of $F_N$. In this paper we prove that for $N\ge 3$ if $H\le Aut(F_N)$ is a subgroup that projects to an infinite normal subgroup in $Out(F_N)$ then the $H$-orbit of an arbitrary nontrivial element $g\in F_N$ is spectrally rigid. We also establish a similar statement for $F_2=F(a,b)$, provided that $g\in F_2$ is not conjugate to a power of $[a,b]$. We also include an appended corrigendum which gives a corrected proof of Lemma 5.1 about the existence of a fully irreducible element in an infinite normal subgroup of of $Out(F_N)$. Our original proof of Lemma 5.1 relied on a subgroup classification result of Handel-Mosher, originally stated by Handel-Mosher for arbitrary subgroups $H\le Out(F_N)$. After our paper was published, it turned out that the proof of the Handel-Mosher subgroup classification theorem needs the assumption that $H$ be finitely generated. The corrigendum provides an alternative proof of Lemma~5.1 which uses the corrected, finitely generated, version of the Handel-Mosher theorem and relies on the 0-acylindricity of the action of $Out(F_N)$ on the free factor complex (due to Bestvina-Mann-Reynolds). A proof of 0-acylindricity is included in the corrigendum.Comment: Included a corrigendum which gives a corrected proof of Lemma 5.1 about the existence of a fully irreducible element in an infinite normal subgroup of of Out(F_N). Note that, because of the arXiv rules, the corrigendum and the original article are amalgamated into a single pdf file, with the corrigendum appearing first, followed by the main body of the original articl

Core and intersection number for group actions on trees

We present the construction of some kind of "convex core" for the product of two actions of a group on \bbR-trees. This geometric construction allows to generalize and unify the intersection number of two curves or of two measured foliations on a surface, Scott's intersection number of splittings, and the apparition of surfaces in Fujiwara-Papasoglu's construction of the JSJ splitting. In particular, this construction allows a topological interpretation of the intersection number analogous to the definition for curves in surfaces. As an application of this construction, we prove that an irreducible automorphism of the free group whose stable and unstable trees are geometric, is actually induced a pseudo-Anosov homeomorphism on a surface

On Minimum Maximal Distance-k Matchings

We study the computational complexity of several problems connected with finding a maximal distance-$k$ matching of minimum cardinality or minimum weight in a given graph. We introduce the class of $k$-equimatchable graphs which is an edge analogue of $k$-equipackable graphs. We prove that the recognition of $k$-equimatchable graphs is co-NP-complete for any fixed $k \ge 2$. We provide a simple characterization for the class of strongly chordal graphs with equal $k$-packing and $k$-domination numbers. We also prove that for any fixed integer $\ell \ge 1$ the problem of finding a minimum weight maximal distance-$2\ell$ matching and the problem of finding a minimum weight $(2 \ell - 1)$-independent dominating set cannot be approximated in polynomial time in chordal graphs within a factor of $\delta \ln |V(G)|$ unless $\mathrm{P} = \mathrm{NP}$, where $\delta$ is a fixed constant (thereby improving the NP-hardness result of Chang for the independent domination case). Finally, we show the NP-hardness of the minimum maximal induced matching and independent dominating set problems in large-girth planar graphs.Comment: 15 pages, 4 figure

Deformation and rigidity of simplicial group actions on trees

We study a notion of deformation for simplicial trees with group actions (G-trees). Here G is a fixed, arbitrary group. Two G-trees are related by a deformation if there is a finite sequence of collapse and expansion moves joining them. We show that this relation on the set of G-trees has several characterizations, in terms of dynamics, coarse geometry, and length functions. Next we study the deformation space of a fixed G-tree X. We show that if X is `strongly slide-free' then it is the unique reduced tree in its deformation space. These methods allow us to extend the rigidity theorem of Bass and Lubotzky to trees that are not locally finite. This yields a unique factorization theorem for certain graphs of groups. We apply the theory to generalized Baumslag-Solitar groups and show that many have canonical decompositions. We also prove a quasi-isometric rigidity theorem for strongly slide-free G-trees.Comment: Published in Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol6/paper8.abs.htm

The square of a block graph

AbstractThe square H2 of a graph H is obtained from H by adding new edges between every two vertices having distance two in H. A block graph is one in which every block is a clique. For the first time, good characterizations and a linear time recognition of squares of block graphs are given in this paper. Our results generalize several previous known results on squares of trees