Journal of Graph Algorithms and Applications the (3,1)-ordering for 4-connected Planar Triangulations

Canonical orderings of planar graphs have frequently been used in graph drawing and other graph algorithms. In this paper we introduce the notion of an (r, s)-canonical order, which unifies many of the existing variants of canonical orderings. We then show that (3, 1)-canonical ordering for 4-connected triangulations always exist; to our knowledge this variant of canonical ordering was not previously known. We use it to give much simpler proofs of two previously known graph drawing results for 4-connected planar triangulations, namely, rectangular duals and rectangle-of-influence drawings.


Background
A canonical ordering of a planar graph is a way of building the graph by iteratively attaching vertices to some "basic graph" (such as an edge) while preserving some connectivity invariant after each iteration. This concept was introduced in the late 1980's by de Fraysseix, Pach and Pollack [dFPP90]. They used the canonical ordering to show that planar graphs can be drawn on a grid of size (2n − 4) × (n − 2). Subsequently, canonical orderings became one of the main tools in graph drawings, e.g. for drawing graphs in grids of small dimensions (see e.g. [dFPP90,CN98]), rectangular duals [KH97], and also graph algorithms such as encoding planar graphs [HKL99] or finding k-disjoint trees in planar graphs [NRN97,NN00].
Our contribution There is now a number of variations of canonical orderings, depending on the connectivity of the graph and whether it is triangulated or not. (We will review these below.) In this paper, we show the existence yet another canonical ordering, this one for planar 4-connected triangulations. It is substantially different from the canonical ordering for such graphs that was presented by Kant and He [KH97]. We call this the (3, 1)-canonical ordering. More generally, we introduce the concept of an (r, s)-canonical ordering, which (roughly speaking) means that the partial graph must be r-connected and the rest-graph must be s-connected; the existing canonical orders all fit into this framework.
We use the (3, 1)-canonical ordering to provide alternate (and, in our opinion, significantly simpler) proofs of two previously known results about 4-connected planar triangulations: they have rectangular duals (Section 4.1) and rectangle-of-influence drawings (Section 4.2).

Review of existing canonical orderings
We assume that the reader is familiar with planar graphs (refer e.g. to [Die12]). We use the term triangulation for a maximal planar simple graph, i.e., a graph in which all faces are triangles and which has 3n − 6 edges of which none is a multiple edge or a loop. Such a graph has a unique planar embedding; we further assume that one face has been fixed as the outer face. We begin our review of canonical ordering with the one for triangulations introduced by de Fraysseix et al. [dFPP90]. We paraphrase their definition to the following one (which is easily shown to be equivalent): Definition 1 (Canonical ordering for triangulations [dFPP90]). Let G be a triangulation with outer face u 1 , u 2 , u 3 . A vertex ordering v 1 , . . . , v n is called a canonical ordering if As we will see later, it will be convenient to define V k := {v k } and so V 1 ∪· · ·∪V n becomes a partition of the vertex set. For any such partition and an index k, we use the notation G k for the subgraph induced by V 1 ∪ · · · ∪ V k and we let the complement G k of G k be the subgraph induced by the vertices V − (V 1 ∪ · · · ∪ V k−1 ). Note that vertex set V k belongs to both G k and G k . 1 One can observe that in a canonical ordering for a triangulation, the complement G k is a connected graph for all k < n. This holds because any vertex v k = u 1 , u 2 , u 3 is not on the outer face and so there must exist some minimal k > k where v k is not on the outer face of G k . Due to the triangular faces, v k receives an edge to v k , and iterating the argument, hence has a path within G k that leads to v n .
We note here, without giving details, that this canonical ordering has been generalized to 3-connected planar graphs that are not necessarily triangulated [Kan96], and also to non-planar 3-connected graphs (see [Sch14] and the references therein).
In 1997, Kant and He [KH97] showed that one can define a different canonical ordering for 4-connected triangulations, and used it to construct visibility representations of 4-connected planar graphs. Its definition, slightly paraphrased, is as follows: Definition 2 (Canonical ordering for 4-connected triangulations [KH97]). Let G be a 4connected triangulation with outer face u 1 , u 2 , u 3 . A vertex order v 1 , . . . , v n is called a canonical ordering for 4-connected triangulations if • For every 1 < k < n, graphs G k and G k are 2-connected.
This canonical ordering was extended to a canonical ordering for all planar 4-connected graphs (not necessarily triangulated) by Nakano, Rahman and Nishizeki [NRN97]. Versions of a canonical order for 4-connected non-planar graphs are also known [CLY05].
Going one higher in connectivity, Nagai and Nakano [NN00] introduced a canonical ordering for 5-connected triangulations. Here, vertices are added in sets that are sometimes more than a singleton. We need a definition. Let G be a graph where all interior faces are triangles. A fan of G is a subset of vertices z 1 , . . . , z f that induces a path with deg(z i ) = 3 for all i = 1, . . . , f . We will only apply this concept if all vertices in the fan belong to the outer face of G. Since interior faces are triangles, it follows that for all z i the third neighbor (i.e., the one not on the outer face) is the same vertex. See also Figure 1(right).
Definition 3 (Canonical ordering for 5-connected triangulations [NN00]). Let G be a 5connected triangulation with outer face u 1 , u 2 , u 3 . A partition of the vertices V = V 1 ∪· · ·∪V L is called a canonical ordering for 5-connected triangulations if • V 2 consists of all neighbors of u 1 and u 2 , • V L−1 consists of all neighbors of u 3 , • For 2 < k < L − 1, vertex set V k is either a single vertex or a fan, • For every 2 < k < L, graph G k is 3-connected and graph G k is 2-connected.
This canonical ordering was used to find 5 independent spanning trees in 5-connected triangulations [NN00]. To our knowledge, it has not been generalized to planar 5-connected (not necessarily triangulated) graphs, and not to non-planar 5-connected graphs either. Since no planar graph is 6-connected, no canonical orderings for higher connectivity can exist for planar graphs.
Note that the three canonical orderings listed here are very similar, with the essence being the connectivity that is required of the subgraphs and their complements. In light of this, we aim to unify the three definitions with the following: Definition 4 ((r, s)-canonical ordering). Let G be a triangulation with outer-face {u 1 , u 2 , u 3 }. We say that a vertex partition V 1 ∪ . . . ∪ V L is an (r, s)-canonical ordering if • u 1 belongs to V 1 and u 3 belongs to V L , and • for every 1 < k < L, graph G k is r-connected and G k is s-connected.
Note that this definition is deliberately vague on the exact form that the vertex sets V k must have. In particular, nothing prevents us (yet) from setting L = 1 and V 1 = V , which satisfies all conditions. The existing canonical orderings restrict V k to be a singleton or, for 5-connected triangulations, fans. Thus the above definition should be viewed as a class of definitions, to be refined further by stating restrictions on the vertex sets V k .
Rephrasing the existing canonical orders in the above terms, the canonical order for triangulations becomes a (2, 1)-canonical ordering with only singletons, the one for 4-connected triangulations becomes a (2, 2)-canonical ordering with only singletons, and the one for 5connected triangulations becomes a (3, 2)-canonical ordering with only singletons or fans. The reader will notice that the sum of the two numbers equals the connectivity of the graph.
Pushing this further, one may ask whether any (r+s)-connected triangulation has an (r, s)canonical ordering such that each V k has some simple form. Note that we may assume that r ≥ s, since a reversal of an (r, s)-canonical ordering gives an (s, r)-canonical ordering. We study here (3, 1)-canonical ordering for 4-connected triangulations, under the restriction that each V k is a singleton or a fan. To our knowledge no such ordering was known before.

(3, 1)-canonical orderings
We have already given the broad idea of a (3, 1)-canonical ordering earlier. We re-state it here and give the specific restrictions imposed on the vertex sets. See also' Figure 1. Figure 1: A singleton V k and a fan V k in a (3, 1)-canonical ordering.
Definition 5. Let G be a 4-connected triangulation with outer-face {u 1 , u 2 , u 3 }. A (3, 1)canonical order with singletons and fans is a partition V = V 1 ∪ · · · ∪ V L such that where z is the third vertex of the interior face adjacent to (u 1 , u 2 ).
• For any 1 < k < L, set V k is either a singleton or a fan.
• For any 1 < k < L, graph G k is 3-connected and G k is connected.
In what follows, we will omit the "with singletons and fans", as we will not study any other version of (3, 1)-canonical orderings. Our main goal is to show that every 4-connected triangulation has such a (3, 1)-canonical ordering. The proof of this proceeds by induction, and we state the crucial lemma for the induction step separately first. We need a few definitions.
A plane graph is called a triangulated disk if every interior face is a triangle and the outerface is a simple cycle. A triangulated disk is called internally 4-connected if its outer-face has no chord, and every triangle is a face. Observe that a triangle is an internally 4-connected triangulated disk, and so is any 4-connected triangulation. Also observe that a subgraph of an internally 4-connected triangulated disk is again an internally 4-connected triangulated disk if and only if its outer-face is a simple cycle that has no chord.
Lemma 1. Let G be an internally 4-connected triangulated disk with n ≥ 4. Let (u 1 , u 2 ) be an edge on the outer-face. Then there exists a vertex set V such that • V contains only outer-face vertices, and none of u 1 , u 2 .
• G − V is an internally 4-connected triangulated disk.
• V is a singleton or a fan.
Proof. 2 Enumerate the outer face vertices of G as u 1 = c 1 , c 2 , . . . , c = u 2 in clockwise order. Define a 2-leg to be a path c i − x − c j where i < j − 1 and x is not on the outer-face. Vertex x is called a 2-leg-center. We always have at least one 2-leg (namely, the one consisting of u 1 = c 1 , u 2 = c and their common neighbor at the interior face incident to (u 1 , u 2 ); this vertex is interior since G has no chord and at least 4 vertices).
We say that a 2-leg-center x dominates a 2-leg-center y if vertex y is strictly inside the cycle x − c i − c i+1 − · · · − c j − x formed by some 2-leg {c i , x, c j } with center-vertex x. See also Figure 2(left). The dominance-relationship is acyclic since any 2-leg with center-vertex y must enclose strictly fewer faces than the 2-leg {c i , x, c j }. Therefore we must have some minimal 2-leg-centers, which are the ones that do not dominate any other 2-leg-center.
By definition for any 2-leg {c i , x, c j }, we have j ≥ i + 2 and so there exists at least one vertex between c i and c j on the outer-face. We say that a 2-leg {c i , x, c j } is basic if the vertices c i+1 , . . . , c j−1 all have degree 3, and complex otherwise. Note that if {c i , x, c j } is basic, then c i+1 , . . . , c j−1 form a fan and their common neighbor is x.
Let x be a minimal 2-leg center. We have two cases: • All 2-legs containing x are basic.
Let i ≥ 1 be minimal and j ≤ be maximal such that x is adjacent to c i and c j . See also Figure 2(middle). Since x is a 2-leg-center, we have i < j − 1. By case assumption the 2-leg {c i , x, c j } is basic, so V = {c i+1 , . . . , c j−1 } is a fan. We verify that G := G − V is an internally 4-connected triangulated disk: -The outer-face of G consists of the one of G plus x. By definition of a 2-center x was not on the outer-face, so G is a triangulated disk. • Some 2-leg {c i , x, c j } is complex.
We assume that i has been chosen maximally, i.e., so that {c i+1 , x, c j } is either not a 2-leg or not complex. We claim that in this case V = {c i+1 } is a suitable vertex set.
We first show that c i+1 cannot be adjacent to x. Assume for contradiction that it is, -If a chord of G connected two vertices in c 1 , . . . , c i , c i+2 , . . . , c , then it would also be a chord in G, which is excluded.
-If a chord connected two non-consecutive vertices in c i =a 0 , . . . , a d+1 =c i+2 , then in G there would be an edge between two non-consecutive neighbors of c i+1 , implying a triangle that is not a face.
-If a chord connected some a s , 1 ≤ s ≤ d, with some c h , i + 2 < h ≤ j, then {c i+1 , a s , c h } would be a 2-leg in G. By minimality of x hence a s = x, but this contradicts that c i+1 is not adjacent to x.
-If a chord connected some a s , 1 ≤ s ≤ d, with some c h , 1 ≤ h < i or j < h ≤ , then by a s = x it would have to cross (c i , x) or (x, c j ), contradicting planarity.
So G is an internally 4-connected triangulated disk.
Observe that in both cases V ⊆ {c i+1 , . . . , c j−1 } for some 1 ≤ i < j ≤ , and so V does not contain u 1 or u 2 as desired.
Proof. We choose the vertex set in reverse order. Let {u 1 , u 2 , u 3 } be the outer-face and choose V L := {u 3 }; this satisfies all conditions since u 3 has at least 3 neighbors. (We do not at this point know the correct value of L, but simply assign indices backwards and shift indices at the end so that the vertex sets are numbered V 1 , . . . , V L .) Observe that G − u 3 is an internally 4-connected triangulated disk, because the neighbors of u 3 form a simple cycle without chord (else there would be a separating triangle at u 3 ). Assume now some V k+1 , . . . , V L have been chosen already such that the remaining graph G k := G − (V k+1 ∪ · · · ∪ V L ) is an internally 4-connected triangulated disk with (u 1 , u 2 ) on the outer-face. If G k has at least 4 vertices, then apply Lemma 1 to find the next V k . Graph G k − V k is again internally 4-connected, so we can continue choosing vertex sets until only 3 vertices, including u 1 and u 2 , are left. Since the graph is still internally 4-connected, these vertices must be a triangle, and hence a face of G. So setting V 1 to be the three vertices of this triangle gives the desired ordering.
To observe that the required connectivity holds, note that any internally 4-connected graph is 3-connected since it is a triangulated disk without a chord. To see that G k is connected, it suffices to show that every vertex except u 3 has a neighbor in a later vertex set; the set of these edges then forms a spanning tree in G k . The argument for this is nearly the same as for (2, 1)-orderings. Clearly each of u 1 , u 2 are adjacent to u 3 . For any vertex z = u 1 , u 2 , u 3 , vertex z is not on the outer face of G, and hence there must exist some minimal k such that z is on the outer face of G k −1 , but not on the outer face of G k . Since faces are triangles, this implies that z is adjacent to some vertex in V k . By the above hence G k is connected for any 1 < k < L.
The proofs of the above results are constructive and lead to polynomial time algorithms. With suitable data structures to keep track of 2-leg-centers, it is not hard to see that a (3, 1)-canonical ordering can be found in linear time; we omit the details.

Applications
In this section, we demonstrate two uses for the (3, 1)-canonical ordering in graph drawing. Both results proved here were known before, but in our opinion the (3, 1)-canonical ordering significantly simplifies the proof of these results.

Rectangular duals
A rectangular dual drawing (or RD-drawing for short) of a planar graph G consists of a set of interior-disjoint rectangles assigned to the vertices of G in such a way that the union of the rectangles forms a rectangle without holes, and the rectangles assigned to vertices v and w touch in a non-zero-length line segment if and only if (v, w) is an edge. The following theorem has been proved repeatedly: Tho84,KH97]). Let G be a 4-connected triangulation, and let e be an edge on the outer-face of G. Then G − e has a rectangular dual.
Previous proofs on this result usually used the (2, 2)-canonical ordering (or some equivalent characterization, such as regular edge labellings). We give here a different proof using the (3, 1)-canonical ordering.
Proof. Let the outer-face be {u 1 , u 2 , u 3 }, chosen such that e = (u 1 , u 2 ). Find a (3, 1)canonical ordering V 1 ∪ · · · ∪ V L of G. We now build the rectangular-dual drawing of G − e by drawing G k − e for k = 1, . . . , L. By construction, e = (u 1 , u 2 ) is an edge on the outerface of G k , and we can hence enumerate the outer-face of G k as c k 1 , . . . , c k k with c k 1 = u 1 and c k k = u 2 . We maintain the invariant that in the RD-drawing of G k , the rectangles of c k 1 , . . . , c k k all attach at the top side of the bounding box, in this order. Such a drawing is easily created for G 1 − e, since G 1 is a triangle and so G 1 − e is a path u 1 − z − u 2 , where z is the third vertex of the interior face at (u 1 , u 2 ). Now assume G k is drawn and consider adding either a singleton or a fan V k+1 . Let a and b be the smallest and largest index such that c k a and c k b are adjacent to a vertex in V k+1 . Extend all rectangles of c k 1 , . . . , c k a and c k b , . . . , c k k upward by one unit. This leaves a "gap" where the rectangles of c k a+1 , . . . , c k b−1 ended. There is at least one such rectangle since b ≥ a + 2 by properties of the (3, 1)-canonical ordering (else G k+1 would not be 3-connected). If V k+1 is a singleton z, then we insert the rectangle for z into this gap. If V k+1 is a fan {z 1 , . . . , z f }, then b = a + 2 and so the gap consists exactly of the top of c k a+1 . Split this range into f pieces and assign rectangles for z 1 , . . . , z f in this place. One easily verifies that this represents all added edges as contacts and satisfies the invariant. So we have the desired RD-drawing.

Rectangle-of-influence drawings
A planar straight-line drawing of a graph is called a (weak, closed) rectangle-of-influence drawing (or RI-drawing for short) if for any edge (u, v) the rectangle R(u, v) defined by u, v is empty, i.e., contains no other points of vertices of the graph. (It may contain parts of other edges.) Here, R(u, v) is the minimum axis-aligned rectangle that contains the points of u and v; it degenerates into a line segment if u or v are on a horizontal or vertical line. The following result is known: ). Let G be a 4-connected triangulation and let e be one edge of the outer-face. Then G − e has a (weak, closed) rectangle-of-influence drawing.
We re-prove this result using the (3, 1)-canonical ordering. We note here that the drawing created is exactly the same as in [BBM99]; the difference lies in that we can find the next vertex set to add much more easily with the (3, 1)-canonical ordering.
Proof. Let the outer-face be {u 1 , u 2 , u 3 }, chosen such that e = (u 1 , u 2 ). Find a (3, 1)canonical ordering V 1 ∪ · · · ∪ V L of G. We now build the RI-drawing of G − e by drawing G k − e for k = 1, . . . , L. By construction e = (u 1 , u 2 ) is an edge on the outer-face of G k , and we can hence enumerate the outer-face of G k as c k 1 , . . . , c k k with c k 1 = u 1 and c k k = u 2 . We maintain the invariant that in the RI-drawing of G k x(c k 1 ) < x(c k 2 ) < · · · < x(c k k ) and y(c k 1 ) > y(c k 2 ) > · · · > y(c k k ). Such a drawing is easily created for G 1 − e, since G 1 is a triangle and so G 1 − e is a path u 1 − z − u 2 , where z is the third vertex of the interior face at (u 1 , u 2 ). Now assume G k is drawn and consider adding either a singleton or a fan V k+1 . Let a be the smallest and b be the largest index such that c k a and c k b are adjacent to a vertex in V k+1 . By 3-connectivity of G k+1 we have b ≥ a + 2. If V k+1 is a singleton z, then define and y(z) = 1 2 y(c k a ) + x(c k a+1 ) .
See also Figure 4(middle). By a ≤ b − 2 adding this new point satisfies the invariant. All rectangles R(z, c k j ) are empty for a ≤ j ≤ b, because they do not intersect the drawing of G k except in rectangles R(c k a , c k a+1 ) and R(c k b−1 , c k b ). So we have the desired RI-drawing. If V k+1 is a fan {z 1 , . . . , z f }, then b = a + 2. For h = 1, . . . , f , define and y(z h ) = f − h + 1 f + 1 y(c k a ) + x(c k a+1 ) .
See also Figure 4(right). By a = b − 2 adding these new points satisfies the invariant. All rectangles R(z h , c k j ) are empty for a ≤ j ≤ b, because they do not intersect the drawing of G k except in rectangles R(c k a , c k a+1 ) and R(c k b−1 , c k b ). So we have the desired RI-drawing.

Conclusion
We showed the existence of new canonical order for 4-connected triangulations. We used this canonical order to give simplified proofs of some previously known graph drawing results for 4-connected triangulations. Furthermore, we provided provided a brief survey of canonical orderings for planar graphs and laid the groundwork for their further investigation. Of particular interest to us are the following questions: • Does every planar c-connected triangulation have an (r, s)-canonical ordering for all r + s = c and reasonable restrictions on vertex sets V k ? The missing case is a (4, 1)canonical ordering for 5-connected triangulations.