Non-leaf Vertices Research Articles

Is this network proper forest-based?

In evolutionary biology, networks are becoming increasingly used to represent evolutionary histories for species that have undergone non-treelike or reticulate evolution. Such networks are essentially directed acyclic graphs with a leaf set that corresponds to a collection of species, and in which non-leaf vertices with indegree 1 correspond to speciation events and vertices with indegree greater than 1 correspond to reticulate events such as gene transfer. Recently forest-based networks have been introduced, which are essentially (multi-rooted) networks that can be formed by adding some arcs to a collection of phylogenetic trees (or phylogenetic forest), where each arc is added in such a way that its ends always lie in two different trees in the forest. In this paper, we consider the complexity of deciding whether a given network is proper forest-based, that is, whether it can be formed by adding arcs to some underlying phylogenetic forest which contains the same number of trees as there are roots in the network. More specifically, we show that it is NP-complete to decide whether a tree-child network with m roots is proper forest-based, for each m≥2. Moreover, for binary networks the problem remains NP-complete when m≥3 but becomes polynomial-time solvable for m=2. We also give a fixed parameter tractable (FPT) algorithm, with parameters the maximum outdegree of a vertex, the number of roots, and the number of indegree 2 vertices, for deciding if a semi-binary network is proper forest-based. A key element in proving our results is a new characterization for when a network with m roots is proper forest-based in terms of certain m-colorings.

K-Planar Placement and Packing of Δ-Regular Caterpillars

This paper studies a packing problem in the so-called beyond-planar setting, that is when the host graph is “almost-planar” in some sense. Precisely, we consider the case that the host graph is [Formula: see text]-planar, i.e., it admits an embedding with at most [Formula: see text] crossings per edge, and focus on families of [Formula: see text]-regular caterpillars, that are caterpillars whose non-leaf vertices have the same degree [Formula: see text]. We study the dependency of [Formula: see text] from the number [Formula: see text] of caterpillars that are packed, both in the case that these caterpillars are all isomorphic to one another (in which case the packing is called placement) and when they are not. We give necessary and sufficient conditions for the placement of [Formula: see text] [Formula: see text]-regular caterpillars and sufficient conditions for the packing of a set of [Formula: see text]-, [Formula: see text]-, [Formula: see text], [Formula: see text]-regular caterpillars such that the degree [Formula: see text] and the degree [Formula: see text] of the non-leaf vertices can differ from one caterpillar to another, for [Formula: see text], [Formula: see text].

Expected Number of Induced Subtrees Shared by Two Independent Copies of a Random Tree

Consider a rooted tree with leaf-set and with all nonleaf vertices having out-degree 2, at least. A rooted tree with leaf-set is induced by in if is the lowest common ancestor subtree for , with all its degree-2 vertices suppressed. A “maximum agreement subtree” (MAST) for a pair of two trees and is a tree with a largest leaf-set such that is induced by both in and . Bryant, McKenzie, and Steel [BioConsensus, AMS, Providence, RI, 2003, pp. 55–65] and Bernstein et al. [SIAM J. Discrete Math., 29 (2015), pp. 2065–2074] proved, among other results, that for and being two independent copies of a random binary (uniform or Yule–Harding distributed) tree , the likely magnitude order of is . We prove this bound for a wide class of random rooted trees: is a terminal tree of a branching, Galton–Watson, process with an ordered-offspring distribution of mean 1, conditioned on “total number of leaves is .”

On the average order of a dominating set of a forest

We show that the average order of a dominating set of a forest graph G on n vertices with no isolated vertices is at most 2n/3. Moreover, the equality is achieved if and only if every non-leaf vertex of G is a support vertex with one or two leaf neighbors. Our result answers an open question of Beaton and Brown in [The average order of dominating sets of a graph, Discrete Math. 344(12) (2021)].

Omnichannel Assortment Optimization Under the Multinomial Logit Model with a Features Tree

Problem definition: We consider the assortment optimization problem of a retailer that operates a physical store and an online store. The products that can be offered are described by their features. Customers purchase among the products that are offered in their preferred store. However, customers who purchase from the online store can first test out products offered in the physical store. These customers revise their preferences for online products based on the features that are shared with the in-store products. The full assortment is offered online, and the goal is to select an assortment for the physical store to maximize the retailer’s total expected revenue. Academic/practical relevance: The physical store’s assortment affects preferences for online products. Unlike traditional assortment optimization, the physical store’s assortment influences revenue from both stores. Methodology: We introduce a features tree to organize products by features. The nonleaf vertices on the tree correspond to features, and the leaf vertices correspond to products. The ancestors of a leaf correspond to features of the product. Customers choose among the products within their store’s assortment according to the multinomial logit model. We consider two settings; either all customers purchase online after viewing products in the physical store, or we have a mix of customers purchasing from each store. Results: When all customers purchase online, we give an efficient algorithm to find the optimal assortment to display in the physical store. With a mix of customers, the problem becomes NP-hard, and we give a fully polynomial-time approximation scheme. We numerically demonstrate that we can closely approximate the case where products have arbitrary combinations of features without a tree structure and that our fully polynomial-time approximation scheme performs remarkably well. Managerial implications: We characterize conditions under which it is optimal to display expensive products with underrated features and expose inexpensive products with overrated features.

Two approaches for the min-degree constrained minimum spanning tree problem

Given an undirected, connected and weighted graph, the min-degree constrained minimum spanning tree (md_MST) problem aims to find a spanning tree (T) with minimum cost subject to each non-leaf vertex in T must have at least d degree, where d is a constant positive integer. Being a NP-hard problem, it finds several practical applications in the design of networks. In this paper, we present two approaches for this problem in which the first approach is a hybrid metaheuristic technique (hABC) combining an artificial bee colony algorithm with local search, and the second approach is iterated local search (ILS). The proposed hABC distinguishes itself from the existing hybrid artificial bee colony algorithm particularly by its size of employed bee population, initial solution generation, neighborhood operators, the way scout bee phase is handled and the local search. The proposed hABC works with a population of only 3 solutions that effectively coordinates with other key components of hABC and helps in finding new high quality solutions particularly for larger size of available benchmark instances. Extensive experiments on four data-sets of 105 benchmark instances demonstrate that ILS and hABC dominate state-of-the-art approaches. Specifically, hABC finds new best values for 41 instances, whereas ILS finds new best values for 28 instances in comparison to state-of-the-art approaches. Experimental results also show that hABC without local search (hABC-LS) is overall superior to state-of-the-art approaches except one where hABC-LS is comparable, but computationally faster on available benchmark instances.

Grundy Domination of Forests and the Strong Product Conjecture

A maximum sequence $S$ of vertices in a graph $G$, so that every vertex in $S$ has a neighbor which is independent, or is itself independent, from all previous vertices in $S$, is called a Grundy dominating sequence. The Grundy domination number, $\gamma_{gr}(G)$, is the length of $S$. We show that for any forest $F$, $\gamma_{gr}(F)=|V(T)|-|\mathcal{P}|$ where $\mathcal{P}$ is a minimum partition of the non-isolate vertices of $F$ into caterpillars in which if two caterpillars of $\mathcal{P}$ have an edge between them in $F$, then such an edge must be incident to a non-leaf vertex in at least one of the caterpillars. We use this result to show the strong product conjecture of B. Brešar, Cs. Bujtás, T. Gologranc, S. Klavžar, G. Košmrlj, B.~Patkós, Zs. Tuza, and M. Vizer, Dominating sequences in grid-like and toroidal graphs, Electron. J. Combin. 23(4): P4.34 (2016), for all forests. Namely, we show that for any forest $G$ and graph $H$, $\gamma_{gr}(G \boxtimes H) = \gamma_{gr}(G) \gamma_{gr}(H)$. We also show that every connected graph $G$ has a spanning tree $T$ so that $\gamma_{gr}(G)\le \gamma_{gr}(T)$ and that every non-complete connected graph contains a Grundy dominating set $S$ so that the induced subgraph of $S$ contains no isolated vertices.

On the extremal values for the Mostar index of trees with given degree sequence

For a given graph G, the Mostar index Mo(G) is the sum of absolute values of the differences between nu(e) and nv(e) over all edges e=uv of G, where nu(e) and nv(e) are, respectively, the number of vertices of G lying closer to u than to v and the number of vertices of G lying closer to v than to u. The degree sequence of a tree is the sequence of the degrees (in descending order) of the non-leaf vertices. This paper determines those trees with a given degree sequence which have the greatest Mostar index. Consequently, all extremal trees with the greatest Mostar index are obtained in the sets of all trees of order n with the maximum degree, the number of leaves, the independence number and the matching number, respectively. On the other hand, some properties of trees with a given degree sequence which have the least Mostar index are given. We also determine those trees with exactly n−3 or n−4 leaves when the degree sequence is given, which have the least Mostar index. At last some numerical results are discussed, in which we calculate the Mostar indices of two sets of molecular graphs: octane isomers and benzenoid hydrocarbons; We compare their Mostar indices with some other distance-based topological indices through their correlations with the chemical properties. The linear model for the Mostar index is better than or as good as the models corresponding to the other distance-based indices.

Formula omitted]-Ary spanning trees contained in tournaments

A rooted tree is called a k-ary tree, if all non-leaf vertices have exactly k children, except possibly one non-leaf vertex has at most k−1 children. Denote by h(k) the minimum integer such that every tournament of order at least h(k) contains a k-ary spanning tree. It is well-known that every tournament contains a Hamiltonian path, which implies that h(1)=1. Lu et al. (1999) proved the existence of h(k), and showed that h(2)=4 and h(3)=8. The exact values of h(k) remain unknown for k≥4. A result of Erdős on the domination number of tournaments implies h(k)=Ω(klogk). In this paper, we prove that h(4)=10 and h(5)≥13.

Bounded degree complexes of forests

Given an arbitrary sequence of non-negative integers λ→=(λ1,…,λn) and a graph G with vertex set {v1,…,vn}, the bounded degree complex, denoted as BDλ→(G), is a simplicial complex whose faces are the subsets H⊆E(G) such that for each i∈{1,…,n}, the degree of vertex vi in the induced subgraph G[H] is at most λi. When λi=k for all i, the bounded degree complex BDλ→(G) is called the k-matching complex, denoted as Mk(G).In this article, we determine the homotopy type of bounded degree complexes of forests. In particular, we show that, for all k≥1, the k-matching complexes of caterpillar graphs are either contractible or homotopy equivalent to a wedge of spheres, thereby proving a conjecture of Vega (2019, Conjecture 7.3). We also give a closed form formula for the homotopy type of the bounded degree complexes of those caterpillar graphs in which every non-leaf vertex is adjacent to at least one leaf vertex.

Reconciling event-labeled gene trees with MUL-trees and species networks.

Phylogenomics commonly aims to construct evolutionary trees from genomic sequence information. One way to approach this problem is to first estimate event-labeled gene trees (i.e., rooted trees whose non-leaf vertices are labeled by speciation or gene duplication events), and to then look for a species tree which can be reconciled with this tree through a reconciliation map between the trees. In practice, however, it can happen that there is no such map from a given event-labeled tree to any species tree. An important situation where this might arise is where the species evolution is better represented by a network instead of a tree. In this paper, we therefore consider the problem of reconciling event-labeled trees with species networks. In particular, we prove that any event-labeled gene tree can be reconciled with some network and that, under certain mild assumptions on the gene tree, the network can even be assumed to be multi-arc free. To prove this result, we show that we can always reconcile the gene tree with some multi-labeled (MUL-)tree, which can then be "folded up" to produce the desired reconciliation and network. In addition, we study the interplay between reconciliation maps from event-labeled gene trees to MUL-trees and networks. Our results could be useful for understanding how genomes have evolved after undergoing complex evolutionary events such as polyploidy.

A hybrid steady-state genetic algorithm for the min-degree constrained minimum spanning tree problem

Given an undirected, connected and edge-weighted graph, the min-degree constrained minimum spanning tree (md-MST) problem aims to find a minimum spanning tree (T) in such a way that each non-leaf vertex in T has degree at least d, where d is given a positive integer constant. This paper proposes a hybrid steady-state genetic algorithm (HSSGA) for the md-MST problem. The proposed HSSGA combines various components – such as problem-specific genetic operators (crossover operator and mutation operator) that are designed in such a way that the min-degree constraint is always maintained, hence resulting into a feasible child solution; and a population updating strategy that helps in maintaining diversity in the population – of the steady-state genetic algorithm with a fast local search. On a set of standard benchmark instances, the proposed HSSGA shows its effectiveness in comparison to state-of-the-art approaches such as four versions of variable neighborhood search, branch and cut algorithm and three versions of generational genetic algorithm. Moreover, HSSGA finds new best values on 32 out of 105 benchmark instances.

A Force-Directed Algorithm for Drawing Directed Graphs Symmetrically

Symmetry is one of the most important aesthetic criteria on graph drawing. It is quite necessary to measure the extent to which the drawings can be considered symmetric. For this purpose, a symmetric metric based on vertex coordinate calculation is proposed in this paper. It is proven theoretically and experimentally that the proposed metric is robust to contraction, expansion, and rotation of drawings. This robustness conforms to human perception of symmetry. Star-subgraphs and cycles are two common structures in digraphs. Both of them have inherent symmetry which should be displayed in drawings. For this purpose, a force-directed algorithm named FDS is proposed which can draw star-subgraphs and cycles as symmetrically as possible. FDS algorithm draws cycles as circles whose positions are fixed to provide a scaffolding for overall layout, renders non-leaf vertices by a standard force-directed layout, and places leaf vertices on concentric circles via a deterministic strategy. A series of experiments are carried out to test FDS algorithm. The results show that FDS algorithm draws digraphs more symmetrically than the existing state-of-the-art algorithms and performs efficiency comparable to O(nlog⁡n) YFHu algorithm.

Combinatorial properties of triplet covers for binary trees

It is a classical result that an unrooted tree T having positive real-valued edge lengths and no vertices of degree two can be reconstructed from the induced distance between each pair of leaves. Moreover, if each non-leaf vertex of T has degree 3 then the number of distance values required is linear in the number of leaves. A canonical candidate for such a set of pairs of leaves in T is the following: for each non-leaf vertex v, choose a leaf in each of the three components of T−v, group these three leaves into three pairs, and take the union of this set over all choices of v. This forms a so-called ‘triplet cover’ for T. In the first part of this paper we answer an open question (from 2012) by showing that the induced leaf-to-leaf distances for any triplet cover for T uniquely determine T and its edge lengths. We then investigate the finer combinatorial properties of triplet covers. In particular, we describe the structure of triplet covers that satisfy one or more of the following properties of being minimal, ‘sparse’, and ‘shellable’.

Recovering normal networks from shortest inter-taxa distance information.

Phylogenetic networks are a type of leaf-labelled, acyclic, directed graph used by biologists to represent the evolutionary history of species whose past includes reticulation events. A phylogenetic network is tree-child if each non-leaf vertex is the parent of a tree vertex or a leaf. Up to a certain equivalence, it has been recently shown that, under two different types of weightings, edge-weighted tree-child networks are determined by their collection of distances between each pair of taxa. However, the size of these collections can be exponential in the size of the taxa set. In this paper, we show that, if we have no "shortcuts", that is, the networks are normal, the same results are obtained with only a quadratic number of inter-taxa distances by using the shortest distance between each pair of taxa. The proofs are constructive and give cubic-time algorithms in the size of the taxa sets for building such weighted networks.

Minimum triplet covers of binary phylogenetic X-trees

Trees with labelled leaves and with all other vertices of degree three play an important role in systematic biology and other areas of classification. A classical combinatorial result ensures that such trees can be uniquely reconstructed from the distances between the leaves (when the edges are given any strictly positive lengths). Moreover, a linear number of these pairwise distance values suffices to determine both the tree and its edge lengths. A natural set of pairs of leaves is provided by any ‘triplet cover’ of the tree (based on the fact that each non-leaf vertex is the median vertex of three leaves). In this paper we describe a number of new results concerning triplet covers of minimum size. In particular, we characterize such covers in terms of an associated graph being a 2-tree. Also, we show that minimum triplet covers are ‘shellable’ and thereby provide a set of pairs for which the inter-leaf distance values will uniquely determine the underlying tree and its associated branch lengths.

A counterexample to Montgomery’s conjecture on dynamic colourings of regular graphs

A dynamic colouring of a graph is a proper colouring in which no neighbourhood of a non-leaf vertex is monochromatic. The dynamic colouring numberχ2(G) of a graph G is the least number of colours needed for a dynamic colouring of G.Montgomery conjectured that χ2(G)≤χ(G)+2 for all regular graphs G, which would significantly improve the best current upper bound χ2(G)≤2χ(G). In this note, however, we show that this last upper bound is sharp by constructing, for every integer n≥2, a regular graph G with χ(G)=n but χ2(G)=2n. In particular, this disproves Montgomery’s conjecture.

Constructing Tree-Child Networks from Distance Matrices

A tree-child network is a phylogenetic network with the property that each non-leaf vertex is the parent of a tree vertex or a leaf. In this paper, we show that a tree-child network on taxa (leaf) set X with an outgroup and a positive real-valued weighting of its edges is essentially determined by the multi-set of all path-length distances between elements in X provided, for each reticulation, the edges directed into it have equal weight. Furthermore, we give a polynomial-time algorithm for reconstructing such a network from this inter-taxa distance information. Such constructions are of central importance in evolutionary biology where phylogenetic networks represent the ancestral history of a collection of present-day taxa.

A strong symmetric formulation for the Min-degree Constrained Minimum Spanning Tree Problem

Given an integer K≥3 and an edge weighted undirected graph G, the Min-Degree Constrained Minimum Spanning Tree Problem (MDMST) asks for a minimum cost spanning tree of G, such that every non-leaf vertex has degree at least K. We introduce a new reformulation for MDMST that consists of splitting spanning trees into two parts: a backbone tree spanning only the non-leaf vertices and an assignment of the leaves to non-leaf vertices. The backbone tree is modelled with a lifted version of generalized subtour breaking constraints (GSECs). Our computational results show that we can go beyond the strongest MDMST Linear Programming lower bounds in the literature. In addition, a Branch-and-cut algorithm based on our new lower bounds seems to be competitive with the best MDMST exact approach, despite the lack of a primal heuristic. As a by product, new best lower bounds are provided for several unsolved MDMST instances.

On the geometric Ramsey numbers of trees

In this paper, we estimate the geometric Ramsey numbers of trees. Given a tree Tn on n vertices and an outerplanar graph Hm on m vertices, we estimate the minimum number M=M(Tn,Hm) such that, for any set of M points in the plane such that no three of them are collinear, and for any 2-colouring of the segments determined by the M points, either the first colour class contains a non-crossing copy of Tn, or the second colour class contains a non-crossing copy of Hm. We prove that M(Tn,Hm)=(n−1)(m−1)+1 if (i) the points are in convex position, Tn is a caterpillar and Hm is Hamiltonian, or if (ii) the points are in general position, Tn has at most two non-leaf vertices, and Hm is Hamiltonian. We also prove several polynomial bounds for M(Tn,Hm) for other classes of Tn and Hm.