Exact-$2$-Relation Graphs

Consider a tree $T$ with leaf set $V$ and a non-negative edge-weight function $\ell:E(T)\to\mathbb{R}^{+}_{0}$. Denote by $\mathcal{P}(x,y)$ the unique path between $x$ and $y$ in $T$. The canonical distance function $d_{T,\ell}:V\times V\to\mathbb{R}^{+}_{0}$ is then defined by

This definition is the starting point for mathematical phylogenetics, which is centered around finite additive metric spaces and their generalizations [1, 2]. It also serves as a basis for defining a large class of graphs that in the recent past has received considerable attention. The pairwise compatibility graphs $G=\mathrm{PCG}(T,\ell,d_{\min},d_{\max})$ has vertex set $V$ and edges

Originally introduced in the context of phylogenetics [3], they have received considerable interest in the last years, see [4, 5] and the references therein, as well as [6, 7, 8]. A further generalization of “multi-interval” PCGs in explored in [9]. In the setting of PCGs and most phylogenetic applications one usually stipulates that $\ell(e)>0$, measuring e.g. the time between two distinct events associated with adjacent vertices of $T$. A class of graphs that is conceptually closely are the exact $k$-leaf powers [10], for which $\lambda(e)=1$ for all edges of $T$ and $d_{\min}=d_{\max}=k$.

In an alternative interpretation, $\ell(e)$ models the number of discrete evolutionary events along an edge of $e$ of $T$. This is of interest in particular in the context of so-called rare genomic changes such as the gene or loss of a particular gene of gene family or a particular genomic rearragement [11]. Some of these convey phylogenetic information that is (nearly) free of homoplasy, i.e., the independent occurrance in independent lineages. Examples of such rare events are the emergence of novel microRNA families [12] or rearrangements of the genomic gene order [13]. Since such events are very unlikely to have occurred more than once in the same manner, they identify phylogenetic groupings that share such an innovation with very little ambiguity. This provides information to resolve also parts the phylogenetic tree where classical, sequence-based methods fail [14]. In this context it is necessary to allow $\ell(e)=0$ because the events of interest are by definition so rare that not all taxa will be distinguished by them. In the same vein it important that events are discrete and hence an integer-valued weight function $\lambda:E(T)\to\mathbb{N}_{0}$. Both conditions on $\ell$ cause subtle but important differences in comparison with the usual definition of PCGs that requires non-zero edge length but otherwise allows arbitrary real values. We will denote these “non-negative integer pairwise compatibility graph” by nniPCG to distinguish them from the better studied class of PCGs with non-zero real-valued edge weights $\lambda$.

The special case in which two leaves $x$ and $y$ of $T$ are separated by a single event, corresponding to graphs of the form $\mathrm{nniPCG}(T,\lambda,1,1)$, was explored in [15] as a model of rare events in evolution. It turns out that this graph class coincides with the forests. The graphs $\mathrm{nniPCG}(T,\lambda,1,\infty)$ requiring at least one event along the path between two leaves also have a very simple structure: they are exactly the complete multipartite graphs [16].

Considering a rooted tree $\vec{T}$ instead of an unrooted tree $T$, it is natural to consider the digraphs with edges $(x,y)$ whenever a certain number of events occured between the last common ancestor $\mathop{lca}(x,y)$ and $y$. This construction appears naturally in the context of horizontal gene transfer (HGT), where one asks whether $\mathop{lca}(x,y)$ and $y$ are separated by at least one HGT event¹¹1HGT refer to the import of gene from an unrelated species e.g. through an infection, ingestion, acquisition via a plasmid. This gives rise to the class of Fitch graphs [17], which form a subclass of di-cographs introduced by [18]. Their underlying undirected graphs are exactly the $\mathrm{nniPCG}(T,\lambda,1,\infty)$, i.e., the complete multipartite graphs.

A related construction requires a certain number of events between $\mathop{lca}(x,y)$ and $y$ and excludes all events between $\mathop{lca}(x,y)$ and $x$. This class of graphs appears naturally when events are directed, i.e., when it is (in general) no possible to revert the effect of an operation in a single step. Probably the best studied type of single genomic events of this type are so-called Tandem-Duplication-Random-Loss events, during which a genomic interval is duplicated and then one of the two copies of each gene is lost at random [19]. The antisymmetric digraphs obtained by single events are characterized in [15].

Our interest in the graphs $\mathrm{nniPCG}(T,\lambda,k,k)$ for general $k\geq 1$ also stems from rare-event phylogenetic data. Since we assume an underlying tree, the distance matrix $d_{T}$ is additive and its entries are small non-negative integers. The fact that all edge lengths $\ell(e)$ are also integers of course imposes additional constraints. As demonstrated e.g. in the context of orthology assignment (a related problem with vertex labeled trees for which the corresponding graphs turn out to be cographs), graph editing can be employed to correct empirically estimated input graphs [20]. This approach requires, however, that constraint on the graphs that can appear are known. In the case of rare-event phylogenetics, we know that the graph with edge set $\{xy|d_{T}(x,y)=k\}$ must be a $\mathrm{nniPCG}(T,\lambda,k,k)$. In the rare-event scenario, the number of pairs of nodes with $d_{T}(x,y)=k$ will quickly decrease with $k$, so that the empirical input graphs will have few edges for larger values of $k$ and thus rarely reveal obstructions. Hence only small value of $k$ are of practical value for detecting measurement errors in the data. Since the $\mathrm{nniPCG}(T,\lambda,1,1)$ are forests, the corresponding graph editing problem amounts to identifying spanning forests, and possible false positive events are edges in cycles. False negatives are not detectable for $k=1$ since there are no non-tree graphs that would become trees by inserting edges. They could be detected, however, as missing edges in the empirical graph for $k=2$ compared to the most similar member of $\mathrm{nniPCG}(T,\lambda,2,2)$. In this contribution, therefore, we are interested in the characterization of the graphs $\mathrm{nniPCG}(T,\lambda,2,2)$, in which edges correspond to exactly two events between two leaves. This graph class is very different from the exact-2-leaf power graphs, which are known to coincide with the disjoint unions of cliques [21, 10]. In contrast, we shall see below that e.g. every path also has a representation as $\mathrm{nniPCG}(T,\lambda,2,2)$.

This contribution is organized as follows: We first consider a few general properties of the slightly more general exactly-$k$-relation $\overset{k}{\sim}$ before investigating for some small graphs and simple graph families whether they can be respresented with respect to the exactly-$2$-relation $\overset{2}{\sim}$ on the leaf set of some tree. Here, we consider the case that $d_{T}(x,y)>0$, i.e., that all leaves are separated by at least one event, and then relax this constraint and characterize the entire graph class $\mathrm{nniPCG}(T,\lambda,2,2)$ for non-negative, integer $\lambda$. Our main result is that these graphs are those whose quotient with respect to the false twin (R-thinness) relation is a block graph. We then consider the oriented version of the problem and give characterization in terms of forbidden subgraphs.

We shall see that the restriction to integer edge weights on the one hand, and the admission of zero-weights on the other hand, make the graphs $\mathrm{nniPCG}(T,\lambda,k,k)$ quite different from the exact-$k$-leaf power graphs studies systematically in [10]. While it is true that every $\mathrm{PCG}(T,\lambda,d_{\min},d_{\max})$ with non-negative real weigths $\lambda$ and bounds $d_{\min}$ and $d_{\max}$ also has a representation as $\mathrm{PCG}(T,\hat{\lambda},\hat{d}_{\min},\hat{d}_{\max})$ with integer weights and bounds [22, Lemma 2], the restriction to integer weights clearly affects the definition of graph classes. For instance, the PCG class with rational weights and bounds $d_{\min}=d_{\max}=1$ contains the $\mathrm{nniPCG}(T,\lambda,k,k)$ for all $k\in\mathbb{N}$. Throughout this contribution we use a notation that is inspired by related work in mathematical phylogenetics.

We consider unrooted instead of rooted tree since the distances $d_{T}(x,y)$ and thus the exactly-$k$-relation $\overset{k}{\sim}$ contains no information on position of root. In fact, it is well known [23, 24] that a metric $d$ of the form (1) uniquely defines an unrooted tree. Therefore, one can only hope to reconstruct the unrooted tree $T$.

In particular, therefore it is sufficient to consider phylogenetic trees, that is, trees $T$ in which every interior node $x\in V(T)\setminus L$ has degree at least $3$. Fig. 1 gives an example. A simple, but important consequence of Definition 1 is the following

It follows that “being explained with respect to the exactly-$k$-relation” is a hereditary graph property for all $k$.

We also note the following immediate consequence of the definition.

It is therefore sufficient to consider connected graphs.

Consider two leaves $x,y\in L$ in an edge-labeled tree $(T,\lambda)$ such that $x\overset{0}{\sim}y$, i.e., $d_{T}(x,y)=0$, and another leaf $z\in L\setminus\{x,y\}$. The triangle inequalities $d_{T}(x,z)\leq d_{T}(x,y)+d_{T}(y,z)$ and $d_{T}(y,z)\leq d_{T}(y,x)+d_{T}(x,z)$ implies $d_{T}(x,z)=d_{T}(y,z)$. Thus $x$ and $y$ have the same neighbors in graph $G$ explained by $T$, i.e., $N_{G}(x)=N_{G}(y)$.

In contrast, true twins, which play no role here, satisfy $N(x)\cup\{x\}=N(y)\cup\{y\}$. By definition, false twins $x\mathrel{\mathrel{\ooalign{\hss\raisebox{-0.73193pt}{$\sim$}\hss\cr\hss\raisebox{3.09999pt}{\scalebox{0.75}{$\bullet$}}\hss}}}y$ are non-adjacent, while true twins are always adjacent [25].

The false twin (R-thinness) relation $\mathrel{\mathrel{\ooalign{\hss\raisebox{-0.73193pt}{$\sim$}\hss\cr\hss\raisebox{3.09999pt}{\scalebox{0.75}{$\bullet$}}\hss}}}$ has been well studied in the literature, in particular in the context of graph products [26]. It is well known that $\mathrel{\mathrel{\ooalign{\hss\raisebox{-0.73193pt}{$\sim$}\hss\cr\hss\raisebox{3.09999pt}{\scalebox{0.75}{$\bullet$}}\hss}}}$ is an equivalence relation, see e.g. [26, sect. 8.2]. Its equivalence classes, which we denote by $R_{i}$, $i=1,\dots,h$, are totally disconnected in $G$ because, by definition, $x\notin N(x)$. Denote by $G[r_{1},r_{2},\dots,r_{h}]$ be the subgraph of $G$ induced by one arbitrarily chosen representative $r_{i}\in R_{i}$ of each false twin class. Since for any $x\in R_{i}$ and $y\in R_{j}$ we have $xy\in E(G)$ if and only $x^{\prime}y^{\prime}\in E(G)$ for all $x^{\prime}\in R_{i}$ and all $y^{\prime}\in R_{j}$ we observe that $G[r_{1},r_{2},\dots,r_{h}]$ and the quotient graph $G/\!\mathrel{\mathrel{\ooalign{\hss\raisebox{-0.73193pt}{$\sim$}\hss\cr\hss\raisebox{3.09999pt}{\scalebox{0.75}{$\bullet$}}\hss}}}$ are isomorphic. An illustration is given in Fig. 2.

A graph is called R-thin [27], point determining graph [28] or mating graph [29] if $\mathrel{\mathrel{\ooalign{\hss\raisebox{-0.73193pt}{$\sim$}\hss\cr\hss\raisebox{3.09999pt}{\scalebox{0.75}{$\bullet$}}\hss}}}$ is discrete, i.e., every false twin class consists of only a single point. Clearly, $G/\!\mathrel{\mathrel{\ooalign{\hss\raisebox{-0.73193pt}{$\sim$}\hss\cr\hss\raisebox{3.09999pt}{\scalebox{0.75}{$\bullet$}}\hss}}}$ is R-thin. R-thin graphs have also been studied from the point of view of combinatorial enumeration [30, 31]. Algorithms for prime-factorization of graphs, furthermore, often operate on $G/\!\mathrel{\mathrel{\ooalign{\hss\raisebox{-0.73193pt}{$\sim$}\hss\cr\hss\raisebox{3.09999pt}{\scalebox{0.75}{$\bullet$}}\hss}}}$, since R-thinness ensures uniqueness of the factorization and allows for highly efficient algorithms [27, 32, 26]. Below we show that it also suffices to consider $G/\!\mathrel{\mathrel{\ooalign{\hss\raisebox{-0.73193pt}{$\sim$}\hss\cr\hss\raisebox{3.09999pt}{\scalebox{0.75}{$\bullet$}}\hss}}}$, i.e., R-thin graphs, in our setting. Indeed, a simple consequence of Lemma 9 is

From here on we will therefore assume the $(T,\lambda)$ is canonical, i.e., it has non-zero labels for all inner edges of $T$. It is important to note, however, that we still need to consider zero weights on the edges incident with leaves. For instance, it not difficult to check that the graph $G/\!\mathrel{\mathrel{\ooalign{\hss\raisebox{-0.73193pt}{$\sim$}\hss\cr\hss\raisebox{3.09999pt}{\scalebox{0.75}{$\bullet$}}\hss}}}$ in Fig. 3, i.e., $K_{3}+e$, cannot be explained by a tree with only non-zero edge weights.

We will first consider the special case of edge labelings with discrete $\overset{0}{\sim}$. In this case every interior vertex of $T$ is incident with at most one zero-weight edge.

The trivial cases $K_{1}$ and $K_{2}$ are explained by the trees $K_{1}$ and $K_{2}$ with label $\lambda(e)=k$ at the unique edge $e$, respectively. For $|L|=3$ there is only a single phylogenetic tree, the star $S_{3}$ with three leaves and two connected graphs, $P_{3}$, and $K_{3}$, see Fig. 4. We denote the edges from the center to leaf $x_{i}$ by $e_{i}$, $1\leq i\leq 3$. Fig. 4 also shows that class of graphs explained w.r.t. $\overset{2}{\sim}$ is much larger than the exact-2-leaf powers graphs, which comprise only the disjoint unions of cliques [21, 10].

The fact that $S_{n}$ is the only “exact-2-leaf root” of $K_{n}$, i.e., the only tree with unit edge weights that explains $K_{n}$ is shown in [10, Lemma 2]. It is not difficult to see that there is also no other choice of non-negative integer labels on $S_{n}$ that explains $K_{n}$:

We note for later reference that the uniqueness results in Lemmas 13 and 14 do not require the precondition that $\overset{0}{\sim}$ is discrete. This observation will be important in the following section.

The fact that $K_{4}-e$ is a forbidden subgraph implies that two cliques in $G$ cannot be “glued together” by a single common edges. It is possible, however, for cliques to touch in a cut vertex as shown by the example of the bowtie graph $B$, which is obtained by gluing together two triangles at a common vertex, see Fig. 4.

Graphs that can be represented as pairwise compatibility graphs of caterpillars have received special attention in the literature [5, 33, 34, 35]. It is not difficult to see that the path $P_{h}$, $h\geq 3$ can represented by a caterpillar in several settings. These results cannot be directly applied in our setting, however. Any two leaves $x$ and $y$ attached to two distinct inner vertices of a caterpillar are separated by at least three edges an thus cannot be in relation $\overset{2}{\sim}$ if we assume strictly positive integer weights. It follows immediately that $P_{h}$ is not an exact-2-leaf power of a caterpillar and that $P_{h}$ cannot be explained by caterpillar unless zero-weights are allowed. An explicit construction in [15] shows that $P_{h}$ is explained by a caterpillar with edge weights in $\{0,1\}$ with respect to exactly-1-relation $\overset{1}{\sim}$. Lemma 4 implies that we can use the same construction to explain $P_{h}$ by a caterpillar with edge weights in $\{0,2\}$, see Fig. 6. It will be important later on that this construction is indeed unique:

Let us now turn to the general case. We first note that all graphs explained w.r.t. $\overset{2}{\sim}$ with discrete $\overset{0}{\sim}$ are chordal, i.e., every cycle of length greater than three has a chord. Even more stringently, every cycle of length $4$ corresponds to a clique in $G$ because the $K_{4}-e$, i.e., the 4-cycle with a chord, is also a forbidden induced subgraph. We note that there is ample literature on the relationship of chordal graphs and PCGs, see e.g. [4, 5]. Due to the differences in the edge weight functions, it is not immediately pertinent to our discussion, however.

A graph $G$ is a block graph [36] if each of its biconnected components is a clique. Lemma 19 thus implies that every graph that can be explained with respect to the exact-$2$-relations with discrete $\overset{0}{\sim}$ is a block graph (see Thm. 21 below for a formal proof). Algorithm 2 (illustrated in Figure 7) explicitly constructs an edge-labeled tree that explains a given block graph.