Semi-Directed Level-2 Networks

• Semi-directed level-2 networks are mixed phylogenetic graphs derived from rooted networks that maintain directed reticulation arcs while undirecting tree-like edges.
• They decompose into blobs with at most two reticulations, allowing reconstruction by using quarnets and displayed quartet methods.
• Their constrained structure facilitates local topology recovery, though extending to higher levels introduces algorithmic and statistical challenges.

Semi-directed level-2 networks are, in current phylogenetic usage, mixed graphs obtained from rooted phylogenetic networks by undirecting all non-reticulation edges and suppressing the root, so that only arcs entering reticulations remain directed. A network is level-$2$ when each blob contains at most two reticulations. This places semi-directed level-2 networks between rooted and unrooted network models: they retain local directional information around reticulation events, but they do not require a globally identified root. In the binary phylogenetic setting, a central result is that semi-directed level-2 networks with at least four leaves are encoded by their quarnets, whereas the corresponding statement fails at level-$3$ (Huber et al., 2024).

1. Formal definition and graph-theoretic setting

A semi-directed network on a leaf set $X$ is obtained from a rooted directed network $N_d$ on $X$ by replacing all arcs by undirected edges except arcs entering reticulations, and then suppressing the root. In the formulation used for semi-directed phylogenetic networks, the rooted partner is a binary rooted phylogenetic LSA network; the semi-directed network is its underlying mixed graph after all non-hybrid edges are undirected and the former root is suppressed. Any rooted directed network giving rise to the same mixed graph is called a rooting of that semi-directed network. A semi-directed network can therefore have multiple rootings (Huber et al., 2024).

The resulting object is unrooted but not undirected. Reticulation arcs encode the local direction of lineage combination events, whereas tree-like regions are left undirected. This representation is motivated by the fact that a global root can be difficult to infer from real data, while the immediate orientation into reticulations may still be identifiable. In the phylogenetic literature summarized here, the main objects are binary semi-directed phylogenetic networks: mixed graphs with labeled leaves, no degree-2 vertices except possibly the root in the directed precursor, and no parallel arcs in the phylogenetic case (Huber et al., 2024).

The internal decomposition is organized by blobs. A blob is the maximal connected subgraph with no cut-edges; equivalently, in the binary setting it is the relevant biconnected component of the underlying undirected graph after suppressing trivial size-$1$ and size-$2$ cases. Contracting each blob to a single vertex and retaining leaves yields the blob tree $B(N)$, the coarse-grained tree-like structure of the network. A network is level-$k$ if each blob contains at most $k$ reticulations; thus level-$3$0 is an unrooted phylogenetic tree, level-$3$1 allows at most one reticulation per blob, and level-$3$2 allows at most two (Huber et al., 2024).

A simple network is one whose non-leaf part consists of a single blob. A strict level-$3$3 network has at least one blob with exactly $3$4 reticulations. In this vocabulary, a semi-directed level-2 network is a semi-directed phylogenetic network in which every blob has at most two reticulations, with simple strict level-2 networks forming the fundamental one-blob case from which the general theory is built (Huber et al., 2024).

2. Internal structure of simple level-2 networks

For simple strict level-$3$5 networks, the principal structural tool is the generator. Given such a network $3$6, its level-2 generator $3$7 is obtained by deleting all leaves and then suppressing degree-2 vertices by the admissible vertex-suppression operations. The generator is again a mixed graph, with directed arcs into reticulations and undirected edges elsewhere. For semi-directed level-2 networks, the generator classification is especially rigid: the semi-directed level-2 generators are exactly the two mixed graphs shown in Figure 1 of the reconstruction paper (Huber et al., 2024).

Every simple strict level-2 network is produced from one of these generators by subdividing edges and arcs, and then attaching leaves to the resulting sides. A side is either an edge or arc of the generator, or a degree-2 reticulation vertex. For an edge or arc side $3$8, the corresponding path $3$9 in the network is the maximal de-suppressed path that collapses to $X$0 when leaves and suppressible vertices are removed. A leaf hangs off a side when it is attached to a vertex on that path, or directly to the reticulation vertex if the side is a degree-2 reticulation side (Huber et al., 2024).

This side decomposition separates two distinct reconstruction questions. The first is which leaves belong to which sides. The second is the order of leaves along each side. Two simple strict level-2 networks are isomorphic up to sides if their generators are isomorphic and the same leaf sets hang off corresponding sides. Quarnets then resolve the remaining ambiguity: if two such networks are isomorphic up to sides and have the same quarnets, then the ordering of leaves along each side is also determined, so the networks are isomorphic (Huber et al., 2024).

The simple level-2 case is therefore highly constrained. There are only two generator types, the side partition is recoverable, and the side order is recoverable. This rigidity is the structural reason that level-$X$1 behaves differently from higher levels.

3. Quarnets and full reconstruction at level-2

The local subnetwork on a leaf subset $X$2 is defined by restriction. One first keeps only vertices lying on $X$3-paths between leaves in $X$4, then applies blob suppression, parallel arc suppression, and repeated vertex suppression until a phylogenetic network on $X$5 is obtained. For semi-directed phylogenetic networks, the restriction $X$6 is again semi-directed phylogenetic. A quarnet is a semi-directed phylogenetic network with exactly four leaves, and the quarnet set of $X$7 is

$X$8

The notion “encoded by quarnets” means that equality of quarnet sets determines the network up to isomorphism fixing leaf labels (Huber et al., 2024).

The level-2 reconstruction theory proceeds in stages. First, simple strict level-2 networks are weakly encoded by quarnets. Second, quarnets determine the blob tree. The key split theorem states that a non-trivial split $X$9 is induced by a cut-edge in the whole network if and only if every quarnet on two leaves from $N_d$0 and two from $N_d$1 exhibits the corresponding $N_d$2-$N_d$3 split. As a consequence, if two semi-directed networks have the same quarnets, then their blob trees are isomorphic (Huber et al., 2024).

These ingredients culminate in the main level-2 theorem: the class of semi-directed, level-2, binary phylogenetic networks with at least four leaves is encoded by quarnets. The proof combines the simple-network result with blob-tree reconstruction and an induction over non-trivial splits. In effect, one first recovers the coarse tree of blobs and then reconstructs each blob from the relevant $N_d$4-leaf restrictions (Huber et al., 2024).

The level boundary is sharp. The same paper proves that semi-directed level-3 binary phylogenetic networks with at least four leaves are not encoded by quarnets: there exist non-isomorphic level-3 networks with identical quarnet sets. The obstruction is the additional internal flexibility of blobs with three reticulations. Level-$N_d$5 has only two generators and tightly controlled side structure; level-$N_d$6 admits enough internal symmetry and redundancy that all $N_d$7-leaf views can coincide even when the global networks do not (Huber et al., 2024).

4. Quartet-based distinguishability and canonical forms

A second, more specialized line of work studies semi-directed level-2 networks through displayed quartet trees rather than full quarnets. The relevant class is the set $N_d$8 of binary, outer-labeled planar, galled, semi-directed level-2 networks, together with the bloblet subclass $N_d$9 consisting of networks with a single non-leaf blob. Here “outer-labeled planar” means that the network admits a planar embedding with all leaves on the unbounded face, and “galled” means that for each hybrid node, the two incoming hybrid edges lie on a cycle containing no other hybrid edges (Holtgrefe et al., 23 Jul 2025).

In this restricted setting, displayed quartets do not in general recover the full network directly; instead they recover a canonical form $X$0. The canonical form is obtained by contracting every 2-blob, contracting every 3-blob, replacing each 4-blob with a representative cycle, replacing each split symmetric 5-blob with its representative 5-cycle, contracting all 3-cycles in blobs, suppressing dts-undetectable 4-cycles, contracting dts-undetectable frontier edges, and undirecting the remaining 4-cycles in sufficiently large blobs. The point of this normalization is to remove precisely those local configurations that displayed quartet trees cannot distinguish (Holtgrefe et al., 23 Jul 2025).

For outer-labeled planar, galled level-2 networks, three data types coincide at the level of distinguishability: canonical form, displayed quartets, and the split system of all displayed trees. More precisely, for two networks $X$1, the conditions $X$2, equality of displayed quartets, and equality of displayed-tree split systems are equivalent. Hence such networks are distinguishable from displayed quartets if and only if their canonical forms differ (Holtgrefe et al., 23 Jul 2025).

The same paper also analyzes an inter-taxon quartet distance of NANUQ type. For level-2 bloblets $X$3, this metric is circular decomposable and has support exactly $X$4, the split system of displayed trees. Consequently, for bloblets, equality of NANUQ metrics is also equivalent to equality of canonical forms. This gives a distance-based route from quartet data to the canonical mixed-graph structure of the blob (Holtgrefe et al., 23 Jul 2025).

This quartet-based theory does not subsume the quarnet result. The quarnet theorem applies to all semi-directed binary level-2 networks and yields full encoding. The displayed-quartet theory applies to the outer-labeled planar, galled subclass and identifies the canonical form rather than every non-canonical detail.

5. Relation to rooted models, general semi-directedness, and terminological variation

Semi-directed level-2 networks stand in close relation to rooted level-2 phylogenetic networks. In the rooted, fully directed setting, recoverable binary level-2 phylogenetic networks are encoded by their trinets, and there is a polynomial-time reconstruction algorithm from trinets with running time $X$5, where $X$6 is the number of input trinets and $X$7 the number of leaves. The same work shows a fundamental obstruction at level-$X$8: distinct level-3 rooted networks can have identical trinet sets (Iersel et al., 2021). The semi-directed theory replaces trinets by quarnets; this shift reflects the fact that removing the global root changes which local subnetworks suffice for reconstruction.

A broader structural characterization of semi-directed and multi-semi-directed networks is available at the mixed-graph level. A mixed graph $X$9 is multi-semi-directed if and only if every vertex satisfies $1$0 and $1$1, $1$2 contains no semi-directed cycle, and $1$3 contains no non-trivial edge-path between two reticulations. For the single-root case, semi-directedness is characterized by the same degree and cycle conditions together with the existence of a $1$4-path between every pair of vertices. In that framework, a semi-directed level-2 network can be regarded as a semi-directed network that admits a rooting which is level-$1$5; the mixed-graph characterization itself is level-independent (Holtgrefe et al., 24 Jul 2025).

The term “semi-directed” also appears outside phylogenetics with a different meaning. In a modified Barabási–Albert model, “limited directedness” refers not to mixed graphs with reticulation arcs, but to an asymmetric update of the Kertész list during preferential attachment. In that model the degree exponent $1$6 decreases from $1$7 toward $1$8 as $1$9 increases, breaking the usual BA universality, and the paper itself does not use the terminology “level-2” (Sumour et al., 2012). This contrast is substantive: in phylogenetics, semi-directedness encodes partial orientation of evolutionary histories; in that network-growth setting, it encodes bias in the attachment mechanism.

6. Algorithmic significance, statistical implications, and open problems

The quarnet theorem has direct identifiability consequences. If a class of semi-directed networks is encoded by quarnets, then any inference pipeline that consistently estimates the correct quarnets can in principle recover the correct network topology. This is one reason the level-2 result is relevant to the statistical consistency of reconstruction programs under development, including the Squirrel software tool (Huber et al., 2024).

The displayed-quartet theory for outer-labeled planar galled level-2 networks gives a complementary route. Because the canonical form is determined by displayed quartets, and because the NANUQ inter-taxon quartet distance is circular decomposable for level-2 bloblets, quartet data such as concordance factors under the Network Multispecies Coalescent can be used to infer the canonical structure through circular split systems and split-network methods such as Neighbor-Net. The same framework suggests a two-stage strategy: reconstruct the tree-of-blobs first, then reconstruct each blob from its quartet-derived split system (Holtgrefe et al., 23 Jul 2025).

What remains incomplete is equally important. The general quarnet paper does not present an explicit polynomial-time algorithm for reconstructing a semi-directed level-2 network from its quarnets, even though the proof is constructive at a high level (Huber et al., 2024). Sequence-length theory is also not yet available at level-$2$0. The current finite-data bounds concern binary level-1 semi-directed networks under the CFN model; extending them beyond level-1 will require additional work because the present closure theory is level-1-specific, higher-level networks are not all outer-labeled planar, and some quartet profiles could contain all three possible quartet trees on four leaves (Frohn et al., 27 Nov 2025).

Move-based search spaces are likewise better understood at level-$2$1 than at level-$2$2. Semi-directed level-1 networks with fixed reticulation number are connected under cut edge transfer, and the full semi-directed level-1 space is connected under the extended move set CET, CET$2$3, and CET$2$4. The same work does not treat level-$2$5; its discussion indicates that a level-2 extension would require new canonical forms and explicit control of two-reticulation blocks (Linz et al., 2023).

Taken together, these results place semi-directed level-2 networks at a precise threshold. They are rich enough that quarnets, rather than trinets or quartets alone, become the decisive local objects in the general theory, yet they remain constrained enough that full reconstruction is still possible. At level-$2$6, that balance is lost: quarnets no longer suffice, and the internal degrees of freedom of blobs exceed what all $2$7-leaf subnetworks can encode (Huber et al., 2024).