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A progressive plane graph, or plane string diagram is an upward planar drawing of an acyclic directed graph (possibly with isolated vertices) in a plane box such that (1) all vertices drawn on the horizontal boundaries are leaves (vertices of degree one); (2) no vertices are drawn on the vertical boundaries, see Fig 2 for an example. It was introduced by Joyal and Street [8, 9] as a core notion in their graphical calculus for monoidal categories, which provides a firm foundation for many explorations of graphical notations in mathematics and physics.

Figure 2: A progressive plane graph or plane string diagram

It is not difficult to see that the notion of an upward plane graph and that of a progressive plane graph are essentially the same thing [7]. A progressive plane graph is just an upward plane graph satisfying the so-called boxed condition which means that (1) the graph is drawn in a plane box; (2) all vertices drawn on the horizontal boundaries are leaves; and (3) no vertices are drawn on the vertical boundaries. To tackle with appearance of isolated vertices, we can change them into isolated virtual edges, see Fig 3 for an example.

=⇒

Figure 3: Change isolated vertices into isolated virtual edges.

3 Deformation of upward planar drawing

Now let us recall Joyal and Street’s definition of a deformation of progressive plane graphs.

Definition 3.1 ([9]). A deformation of two progressive plane graphs Γ1, Γ2 is a planar isotopy connecting Γ1 and Γ2 such that each intermediate plane graph is a progressive plane graph.

2 There are sufficient algebraic reasons for this definition, which lies in the axioms of monoidal categories and the formalism of string diagrams (we will not give a detailed exposition here, but refer to [9, 12, 7, 4] and Sect.2 of [14] for readers). The crucial property of this notion — each intermediate plane graph is a progressive plane graph — is called recumbent. Easily, we can generalize this notion for upward plane graphs.

Definition 3.2. A deformation of two upward plane graphs G1, G2 is a planar isotopy con- necting G1 and G2 such that each intermediate plane graph is an upward plane graph.

In other words, we say that a deformation of two upward plane graphs is a recumbent planar isotopy connecting them. Similarly, we have the following notion, which is called an upward planar morph in [10].

Definition 3.3. A deformation of two upward planar drawings is a planar isotopy connecting them such that each intermediate planar drawing is upward.

Clearly, these notions of deformations define equivalence relations for progressive plane graphs, upward plane graphs and upward planar drawings, and these equivalence relations have strong algebraic motivations and provide a partial answer to (Q1), only from an algebraic point of view. These deformations are not plain planar isotopies, which defines equivalence relation for plane graphs, but special planar isotopies with a strong constraint — the recumbency property. Then there is a question here: (Q3) in what sense can we say that upward planarity is a topological property? For a complete answer to (Q1), we should answer (Q3), which means to justify the above notions of deformations from a topological view of point.

4 Bimodal rotation and polarization structure

A basic result in theory [1, 11] is the rotation principle (due to Heffter, Edmonds and Ringel), which says that the equivalence class of an planar drawing of a graph on the plane or an surface can be characterized by a rotation system on this graph. A rotation of a vertex v of a graph is a cyclic order y on the set E(v) of its incident edges, denoted as σv. A rotation system σ of graph G is a collection of rotations, one for each vertex, and we write σ = {σv|v ∈ V (G)}. For directed graphs, a similar notion should be bimodal rotation system. For a vertex v of a directed graph, a natural notion is that of a bimodal rotation [6], which is a rotation of v such that both the set I(v) of incoming edges and the set O(v) of outgoing edges are intervals with respect to the cyclic order, see Fig 4 for examples and a non-example. Note that for a source or a sink, any of its rotations is naturally bimodal.

Proposition 4.1 ([13]). Any upward planar graph induces a unique bimodal rotation system on its underlying acyclic directed graph.

yes yes yes no

Figure 4: Bimodal rotations and a non-bimodal rotation

In the theory of graphical calculi, the similar notion is that of a polarization structure. A polarization [9] of a vertex v of a directed graph is a choice of a linear order on the set I(v) of incoming edges and a linear order on the set O(v) of outgoing edges, see Fig 5 for an example. A

3 2 · · · m 1 1 2 k · · ·

1 2 · · · n · · · 1 2 l

Figure 5: Examples of polarizations. polarization structure of a directed graph is a collection of polarizations of all its vertices. The notion of a polarization is same as that of 2-linear lists of edges in [2, 3]. Bimodal rotation has close relations with polarization. As shown in Fig 6, any polarization of v induces a bimodal rotation y in the way that (1) for any f ∈ O(v), f y h1 y h2 ⇐⇒ h1 < h2 in I(v); (2) for any h ∈ I(v), h y f1 y f2 ⇐⇒ f2 < f1 in O(v). Conversely, if v is neither a source nor a sink, then any bimodal rotation of v can define a polarization of v in the above way. However, for a source or a sink, there is no canonical way to define a polarization from a (bimodal) rotation, and the reason is that to break a cyclic order into a linear order we have to choice an element as the first (or last) element and in this case there is no such a natural choice.

Definition 4.2 ([7]). A vertex of a directed graph is called processive if it is neither a source and a sink.

I(v) 1 2 · · · k

v

1 2 · · · l O(v)

Figure 6: A polarization and its equivalent bimodal rotation of a processive vertex.

The following lemma is clear.

Lemma 4.3. For a processive vertices or a leaf, its bimodal rotations and polarizations are in bijective with each other.

Just as the case of progressive plane graphs, we have the following result.

Proposition 4.4 ([9]). Any upward plane graph induces a unique polarization structure on the underlying acyclic directed graph.

5 NPP-extension and main result

First, let us recall a notion.

Definition 5.1 ([7]). A processive graph is a non-empty acyclic directed graph with all sources and sinks being leaves (vertices of degree one).

A processive graph is a non-empty acyclic directed graph with non-leaves being processive. By Lemma 4.3, the bimodal rotation systems of a progressive graph are in bijective with its polarization structures .

4 Definition 5.2. For an acyclic directed graph G, its normal processive-extension (or NP- b extension for short) is the processive graph G obtained from G by the following two kinds of procedures (see Fig. 7): (1) for each source s of G, add one leaf ls and one directed edge es such that O(ls)= {es} = I(s); (2) for each sink t of G, add one leaf lt and one directed edge et such that O(t)= {et} = I(lt). For any acyclic directed graph, its NP-extension exists and is unique, which is just a processive graph obtained by adding an input edge for each source and adding an output edge for each sink.

ls · · · es t s et · · · lt

Figure 7: Normal processive-extensions of s and t.

For upward plane graphs, we have a similar notion. Definition 5.3. Let Γ be an upward plane graph, its normal processive planar extension b (or NPP-extension) is the upward plane graph Γ obtained from Γ by adding an input edge for each source and adding an output edge for each sink. Fig 8 shows an example of a NPP-extension. The notion of a NPP-extension is equivalent to that of a upward consistent assignment [2, 3], which can be explained by the unit convention in graphical calculus for monoidal categories. Lemma 5.4. For an upward plane graph, its NPP-extension is always exists and is unique.

Figure 8: An example of NPP extension/upward consistent assignment

By Lemma 4.3 ,Lemma 5.4 and Proposition 4.4 , we can easily get the following theorem. Theorem 5.5. For an upward plane graph, its NPP-extension and the polarization structure of the underlying acyclic directed graph are equivlent. The following theorem is a direct consequence of the combinatorial characterization of upward planar drawings by Bertolazzi, etc [2, 3]. Theorem 5.6. Two upward plane graphs are related by a deformation if and only if their NPP- extensions are related by a plane isotopy. Combine Theorem 5.5 and Theorem 5.6, we get the following main result.

5 Theorem 5.7. Two upward planar graphs are related by a deformation if and only if they are connected by a planar isotopy which preserves the orientation and polarization of the underlying acyclic directed graph.

This theorem implies that two upward planar drawings of an acyclic directed graph G are equivalent (connected by a deformation) if and only if they are connected by a planar isotopy which preserves the orientation and polarization of G. This theorem also gives a positive answer to Selinger’s conjectue [12] with the strategy different from that of Delpeuch and Vicary [5].

References

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[2] P. Bertolazzi, G. Di Battista. On upward drawing testing of triconnected digraphs. Proc. 7th Annu. ACM Sympos. Comput. Geom., 272-280, 1991.

[3] P. Bertolazzi, G. Di Battista, G. Liotta, and C.Mannino. Upward drawings of triconnected digraphs. Algorithmica, 12:476–497, 1994.

[4] J.C. Baez and M. Stay. Physics, topology, logic and computation: a Rosetta Stone. In: Coecke B. (eds) New Structures for Physics. Lecture Notes in Physics, vol 813. Springer, Berlin, Heidelberg, 2010.

[5] A. Delpeuch and J. Vicary. Normalization for planar string diagrams and a quadratic equiv- alence algorithm. arXiv:1804.07832.

[6] A. Garg and R. Tamassia. Upward . Order, 12(2):109–133, 1995.

[7] S. Hu, X. Lu and Y. Ye. A graphical calculus for semi-groupal categories. Applied Categorical Structures, 27(2):163–197, April 2019, arXiv:1604.07276.

[8] A. Joyal and R. Street. Planar diagrams and tensor algebra. Unpublished manuscript, 1988.

[9] A. Joyal and R. Street. The geometry of tensor calculus. I. Adv. Math., 88(1):55–112, 1991.

[10] G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani and V. Roselli. Upward Planar Morphs. In: T.Biedl, A.Kerren (eds) and Network Visualization. GD 2018. Lecture Notes in Computer Science, vol 11282. Springer, Cham.

[11] B. Mohar and C. Thomassen. Graphs on surfaces. Johns Hopkins series in the mathematical sciences, 2001.

[12] P. Selinger. A survey of graphical languages for monoidal categories. New structures for physics, Lecture Notes in Physics, Springer, 813:289-355, 2011.

[13] R. Tamassia and I. G. Tollis. A unified approach to visibility representations of planar graphs. Discrete Comput. Geom., 1(4):321–341, 1986.

[14] C. Wingfield. Graphical foundations for diagogue games. Ph.D. thesis, 2013.

Xuexing Lu Email: [email protected]

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