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Yangjing Long

Publications and source records attributed to Yangjing Long.

At least 19 recordsLinked to original sources
arXiv2026

Proximity Measures for Classes of Phylogenetic Networks

Phylogenetic networks are used to represent the evolutionary history of species. Due to biological interpretations and computational advantages, researchers have focused on restricted classes of phylogenetic networks, such as tree-child, orchard, and tree-based. These classes capture different notions of tree-likeness: tree-child networks require every internal vertex to have a taxon reachable by a tree path, orchard networks are trees with horizontal arcs (for modelling histories rife with horizontal gene transfers), and tree-based networks are trees with additional (not-necessarily horizontal) arcs. A natural question to ask is ``how far is a given network from belonging to a particular class?'' This motivates the study of proximity measures, which measure the minimum number of graph modifications required to transform a network into one belonging to a particular class. In this paper, we consider three proximity measures based on leaf addition, valid arc deletion, and arc deletion. We study pairwise comparability of the proximity measures, prove complexity results, and derive extremal bounds for the classes of tree, tree-child, orchard, and tree-based networks.

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arXiv2026

Constrained homomorphism orders

We study partial orders induced by constrained variants of finite graph homomorphisms: monomorphisms, embeddings, full homomorphisms, vertex-surjective, edge-surjective and surjective homomorphisms, and locally injective, locally surjective and locally bijective homomorphisms. For each order we ask for analogues of the standard structural properties of the graph homomorphism order: canonical cores, past- or future-finiteness, universality, gaps and finite dualities. The comparison shows which phenomena are specific to ordinary homomorphisms and which are consequences of simpler order-theoretic mechanisms. We identify cores for full and surjective homomorphisms, relate full-homomorphism cores to point-determining graphs, characterize gaps in the full homomorphism order, and give finite obstruction bounds for several one-sided finite orders. We also analyze locally constrained homomorphisms on connected graphs. In particular, locally injective homomorphisms have all connected graphs as cores, admit infinite-chain density under natural degree-refinement assumptions, have explicit gap witnesses, and are universal already on finite connected bipartite subcubic cactus graphs. The paper reorganizes and extends several earlier arguments into a single framework for constrained homomorphism orders.

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arXiv2026

An Explicit $O(r\log r)$ Threshold for Attaining the Semple--Steel Bound with $r$-State Characters

Let $d_r(n)$ be the maximum, over all binary phylogenetic trees with $n$ leaves, of the minimum number of $r$-state characters required to define the tree. Semple and Steel proved that $d_r(n)\geq\lceil(n-3)/(r-1)\rceil$, and Bordewich and Semple proved that equality holds for each fixed $r$ and all sufficiently large $n$. We study the corresponding threshold $n_r$, the least $N$ for which equality holds for every $n\geq N$. The Bordewich--Semple construction yields an explicit polynomial upper bound of order $O(r^5)$ for this threshold. We prove the near-linear estimate \[ 3r+1\leq n_r\leq \ceil{64(r-1)\log_2(r+1)}+3\qquad(r\geq4). \] The proof constructs, for every binary phylogenetic tree with $m=n-3$ internal edges, a linked quartet certificate whose conflict graph has maximum degree at most $16\lceil\log_2(m+2)\rceil+4$. Equitable coloring then packs the certificate into exactly $\lceil m/(r-1)\rceil$ $r$-state characters once $m\geq\lceil64(r-1)\log_2(r+1)\rceil$. We also include the lower bound $n_r\geq3r+1$, obtained from the snowflake obstruction, and state the natural conjecture that this lower bound is the exact threshold for all $r\geq4$. The conjectural endpoint is consistent with the known small-state thresholds: $n_4=13$ and $n_5=16$, while the cases $r=2,3$ are also explicitly classified.

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arXiv2026

Exact Leaf Powers on Cycles, Ladders, Crowns, and Multipartite Block Graphs

Exact \(k\)-leaf powers are graphs whose edges are exactly the pairs of leaves at distance \(k\) in a tree. We prove explicit structure theorems for exact leaf powers on several representative graph families that test different exact-distance phenomena. Our most detailed root-classification theorem concerns chordless cycles: all exact \(5\)-leaf roots of \(C_\ell\), \(\ell\ge 8\), are described by a complete terminal block language. We also prove that the \(t\)-square ladder \(L_t\) is an exact \(5\)-leaf power if and only if \(t\le 2\). In contrast, dense bipartite square structures are often representable: among block-complete multipartite graphs, the exact \(5\)-leaf powers are precisely the bipartite members, and every bipartite co-cluster graph, including every crown \(K_{n,n}-M\), is an exact \(k\)-leaf power for every \(k\ge 5\). Finally, we give parity classifications for complete multipartite graphs and multipartite block graphs at larger exact distances, and isolate a sharp fan boundary at exact distance six.

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arXiv2026

Minimum Network Level Forced by Hardwired Cluster Data

Reticulate evolutionary events, such as hybridization, recombination, and horizontal transfer, can make a tree model inadequate. When evolutionary data are summarized as hardwired clusters, one can ask how much local reticulation complexity is forced by the data itself. We address this question for an arbitrary cluster system $\mathcal C$ on a finite taxon set $X$ by computing the minimum level of a rooted phylogenetic network whose hardwired cluster system is exactly $\mathcal C$. Writing $H=\mathcal H[\mathcal C]$, we define for each non-trivial block $B$ of $H$ a parameter $\mu(B)$ from generating sets of incompatibility intersections in $B$. If $\ell(\mathcal C)$ denotes the minimum level of any rooted network $N$ with $C_N=\mathcal C$, then \[ \ell(\mathcal C)=\max\{\,\mu(B)\mid B\text{ is a non-trivial block of }H\,\}. \] Equivalently, $\mathcal C$ is realizable by a rooted level-$k$ network if and only if $\mu(B)\le k$ for every non-trivial block $B$ of $H$. The lower-bound proof relates incompatibility intersections to non-root hybrid vertices in realizing blocks, while the upper-bound proof starts from the Hasse diagram and iteratively splits selected hybrid vertices without changing the hardwired cluster system. The result turns a network-design problem into a cluster-side criterion and provides an interpretable complexity score for hardwired cluster data, distinct from softwired cluster representation where clusters need only occur in one displayed tree.

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arXiv2026

The Quantum Homomorphism Orders are Universal

Quantum graph homomorphisms, introduced by Man\v{c}inska and Roberson, form a natural quantum relaxation of classical graph homomorphisms. Since this relaxation may create new comparabilities, it could in principle collapse antichains and other order-theoretic configurations. We prove that this does not happen: the quantum homomorphism quasi-order of finite directed graphs is countably universal, and the quantum homomorphism quasi-order of finite planar graphs of maximum degree at most $7$ is also countably universal. Consequently, the same universality holds for finite undirected graphs and for the corresponding quotient partial orders. The result is constructive for finite patterns. Given any finite poset $P$, we explicitly construct finite planar graphs $G_p$, $p\in P$, with $\Delta(G_p)\le 7$, such that \[ p\le_P q \quad\Longleftrightarrow\quad G_p\toq G_q. \] For directed graphs, the proof uses disjoint unions of clockwise directed cycles, where quantum and classical homomorphisms coincide. For undirected graphs, the main ingredient is a finite ordered indicator whose terminals are quantum endpoint-forcing. This gives a quantum analogue of the classical ordered-indicator method: the classical endpoint-image condition is replaced by the projection-level vanishing condition \[ F_{a,p}F_{b,q}=0 \] for every illegal ordered terminal pair. The fixed indicator encodes directed-cycle constructions inside planar bounded-degree graphs without creating extra quantum homomorphisms.

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arXiv2026

Computing the Arc-Deletion Distance to Orchard Networks is NP-hard

Phylogenetic networks generalize phylogenetic trees by allowing reticulate evolutionary events such as horizontal gene transfer and hybridization. Among the many subclasses of phylogenetic networks, orchard networks have attracted increasing attention due to their structural and algorithmic properties. In this paper, we study the arc-deletion distance to orchard networks, defined as the minimum number of reticulate arcs whose deletion transforms a phylogenetic network into an orchard network. We prove that computing this distance is NP-hard via a polynomial-time reduction from the Degree-3 Vertex Cover problem. Our result establishes the computational intractability of this proximity measure and contributes to the complexity theory of phylogenetic network transformations.

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arXiv2025

Defining a phylogenetic tree with the minimum number of small-state characters

Phylogenetic trees represent evolutionary relationships and can be uniquely defined by sets of finite-state biological characteristics. Despite prior work showing that sufficiently large trees can be determined by $r$-state character sets, the minimal leaf thresholds $n_r$ remain largely unknown. In this work, we establish the 3-state case as $n_3 = 8$, providing a concrete base for higher-state analyses. We then resolve the 5-state problem by constructing a counterexample for $n=15$ and proving that for $n \geq 16$, $\lceil (n-3)/4 \rceil$ 5-state characters suffice to uniquely define any tree. Our approach relies on rigorous mathematical induction with complete verification of base cases and logically consistent inductive steps, offering new insights into the minimal conditions for character-based tree identification.

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arXiv2020

Exact-$2$-Relation Graphs

Pairwise compatibility graphs (PCGs) with non-negative integer edge weights recently have been used to describe rare evolutionary events and scenarios with horizontal gene transfer. Here we consider the case that vertices are separated by exactly two discrete events: Given a tree $T$ with leaf set $L$ and edge-weights $λ: E(T)\to\mathbb{N}_0$, the non-negative integer pairwise compatibility graph $\textrm{nniPCG}(T,λ,2,2)$ has vertex set $L$ and $xy$ is an edge whenever the sum of the non-negative integer weights along the unique path from $x$ to $y$ in $T$ equals $2$. A graph $G$ has a representation as $\textrm{nniPCG}(T,λ,2,2)$ if and only if its point-determining quotient $G/\!\rthin$ is a block graph, where two vertices are in relation $\rthin$ if they have the same neighborhood in $G$. If $G$ is of this type, a labeled tree $(T,λ)$ explaining $G$ can be constructed efficiently. In addition, we consider an oriented version of this class of graphs.

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arXiv2020

Unrooted non-binary tree-based phylogenetic networks

Phylogenetic networks are a generalization of phylogenetic trees allowing for the representation of non-treelike evolutionary events such as hybridization. Typically, such networks have been analyzed based on their `level', i.e. based on the complexity of their 2-edge-connected components. However, recently the question of how `treelike' a phylogenetic network is has become the center of attention in various studies. This led to the introduction of \emph{tree-based networks}, i.e. networks that can be constructed from a phylogenetic tree, called the \emph{base tree}, by adding additional edges. While the concept of tree-basedness was originally introduced for rooted phylogenetic networks, it has recently also been considered for unrooted networks. In the present study, we compare and contrast findings obtained for unrooted \emph{binary} tree-based networks to unrooted \emph{non-binary} networks. In particular, while it is known that up to level 4 all unrooted binary networks are tree-based, we show that in the case of non-binary networks, this result only holds up to level 3.

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arXiv2019

Classes of treebased networks

Recently, so-called treebased phylogenetic networks have gained considerable interest in the literature, where a treebased network is a network that can be constructed from a phylogenetic tree, called the base tree, by adding additional edges. The main aim of this manuscript is to provide some sufficient criteria for treebasedness by reducing phylogenetic networks to related graph structures. While it is generally known that deciding whether a network is treebased is NP-complete, one of these criteria, namely edgebasedness, can be verified in linear time. Surprisingly, the class of edgebased networks is closely related to a well-known family of graphs, namely the class of generalized series parallel graphs, and we will explore this relationship in full detail. Additionally, we introduce further classes of treebased networks and analyze their relationships.

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arXiv2018

Reconstructing unrooted phylogenetic trees from symbolic ternary metrics

In 1998, Böcker and Dress gave a 1-to-1 correspondence between symbolically dated rooted trees and symbolic ultrametrics. We consider the corresponding problem for unrooted trees. More precisely, given a tree $T$ with leaf set $X$ and a proper vertex colouring of its interior vertices, we can map every triple of three different leaves to the colour of its median vertex. We characterise all ternary maps that can be obtained in this way in terms of 4- and 5-point conditions, and we show that the corresponding tree and its colouring can be reconstructed from a ternary map that satisfies those conditions. Further, we give an additional condition that characterises whether the tree is binary, and we describe an algorithm that reconstructs general trees in a bottom-up fashion.

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arXiv2018

Phylogenetic trees and homomorphisms

In Chapter 1 we fully characterise pairs of finite graphs which form a gap in the full homomorphism order. This leads to a simple proof of the existence of generalised duality pairs. We also discuss how such results can be carried to relational structures with unary and binary relations. In Chapter 2 we show a very simple and versatile argument based on divisibility which immediately yields the universality of the homomorphism order of directed graphs and discuss three applications. In chapter 3, we show that every interval in the homomorphism order of finite undirected graphs is either universal or a gap. Together with density and universality this "fractal" property contributes to the spectacular properties of the homomorphism order. In Chapter 4 we analyze the phylogenetic information content from a combinatorial point of view by considering the binary relation on the set of taxa defined by the existence of a single event separating two taxa. We show that the graph-representation of this relation must be a tree. Moreover, we characterize completely the relationship between the tree of such relations and the underlying phylogenetic tree.

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arXiv2017

A Short Note on Undirected Fitch Graphs

The symmetric version of Fitch's xenology relation coincides with class of complete multipartite graph and thus cannot convey any non-trivial phylogenetic information.

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arXiv2017

Word-representability of split graphs

Letters $x$ and $y$ alternate in a word $w$ if after deleting in $w$ all letters but the copies of $x$ and $y$ we either obtain a word $xyxy\cdots$ (of even or odd length) or a word $yxyx\cdots$ (of even or odd length). A graph $G=(V,E)$ is word-representable if and only if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy\in E$. It is known that a graph is word-representable if and only if it admits a certain orientation called semi-transitive orientation. Word-representable graphs generalize several important classes of graphs such as $3$-colorable graphs, circle graphs, and comparability graphs. There is a long line of research in the literature dedicated to word-representable graphs. However, almost nothing is known on word-representability of split graphs, that is, graphs in which the vertices can be partitioned into a clique and an independent set. In this paper, we shed a light to this direction. In particular, we characterize in terms of forbidden subgraphs word-representable split graphs in which vertices in the independent set are of degree at most 2, or the size of the clique is 4. Moreover, we give necessary and sufficient conditions for an orientation of a split graph to be semi-transitive.

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arXiv2017

Inference of Phylogenetic Trees from the Knowledge of Rare Evolutionary Events

Rare events have played an increasing role in molecular phylogenetics as potentially homoplasy-poor characters.In this contribution we analyze the phylogenetic information content from a combinatorial point of view by consid-ering the binary relation on the set of taxa defined by the existence of a single event separating two taxa. We showthat the graph-representation of this relation must be a tree. Moreover, we characterize completely the relationshipbetween the tree of such relations and the underlying phylogenetic tree. With directed operations such as tandem-duplication-random-loss events in mind we demonstrate how non-symmetric information constrains the position ofthe root in the partially reconstructed phylogeny.

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arXiv2017

Fractal property of the graph homomorphism order

We show that every interval in the homomorphism order of finite undirected graphs is either universal or a gap. Together with density and universality this "fractal" property contributes to the spectacular properties of the homomorphism order. We first show the fractal property by using Sparse Incomparability Lemma and then by more involved elementary argument.

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