The -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 -canonical order, which unifies many of the existing variants of canonical orderings. We then show that -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.
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 . 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 -disjoint trees in planar graphs [NRN97, NN00].
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 -canonical ordering. More generally, we introduce the concept of an -canonical ordering, which (roughly speaking) means that the partial graph must be -connected and the rest-graph must be -connected; the existing canonical orders all fit into this framework.
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 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 be a triangulation with outer face . A vertex ordering is called a canonical ordering if
, , ,
For every the subgraph of induced by vertices is -connected.
As we will see later, it will be convenient to define and so becomes a partition
of the vertex set. For any such partition and an index , we use the notation for the subgraph induced by
and we let the complement
of be the subgraph induced by the vertices .
Note that vertex set belongs to both and .
One can observe that in a canonical ordering for a triangulation, the complement is a connected graph for all . This holds because any vertex is not on the outer face and so there must exist some minimal where is not on the outer face of . Due to the triangular faces, receives an edge to , and iterating the argument, hence has a path within that leads to .
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 be a 4-connected triangulation with outer face . A vertex order is called a canonical ordering for 4-connected triangulations if
, , ,
For every , graphs and are -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 be a graph where all interior faces are triangles. A fan of is a subset of vertices that induces a path with for all . We will only apply this concept if all vertices in the fan belong to the outer face of . Since interior faces are triangles, it follows that for all 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 be a 5-connected triangulation with outer face . A partition of the vertices is called a canonical ordering for 5-connected triangulations if
consists of all neighbors of and ,
consists of all neighbors of ,
For , vertex set is either a single vertex or a fan,
For every , graph is -connected and graph is -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 (-canonical ordering).
Let be a triangulation with outer-face . We say that a vertex partition is an -canonical ordering if
belongs to and belongs to , and
for every , graph is -connected and is -connected.
Note that this definition is deliberately vague on the exact form that the vertex sets must have. In particular, nothing prevents us (yet) from setting and , which satisfies all conditions. The existing canonical orderings restrict to be a singleton or, for -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 .
Rephrasing the existing canonical orders in the above terms, the canonical order for triangulations becomes a -canonical ordering with only singletons, the one for 4-connected triangulations becomes a -canonical ordering with only singletons, and the one for 5-connected triangulations becomes a -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 -connected triangulation has an -canonical ordering such that each has some simple form. Note that we may assume that , since a reversal of an -canonical ordering gives an -canonical ordering. We study here -canonical ordering for 4-connected triangulations, under the restriction that each is a singleton or a fan. To our knowledge no such ordering was known before.
3 -canonical orderings
We have already given the broad idea of a -canonical ordering earlier. We re-state it here and give the specific restrictions imposed on the vertex sets. See also‘Figure 1.
Let be a 4-connected triangulation with outer-face . A -canonical order with singletons and fans is a partition such that
, where is the third vertex of the interior face adjacent to .
For any , set is either a singleton or a fan.
For any , graph is -connected and is connected.
In what follows, we will omit the “with singletons and fans”, as we will not study any other version of -canonical orderings. Our main goal is to show that every 4-connected triangulation has such a -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 outer-face 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.
Let be an internally -connected triangulated disk with . Let be an edge on the outer-face. Then there exists a vertex set such that
contains only outer-face vertices, and none of .
is an internally -connected triangulated disk.
is a singleton or a fan.
Enumerate the outer face vertices of as in clockwise order. Define a 2-leg to be a path where and is not on the outer-face. Vertex is called a 2-leg-center. We always have at least one 2-leg (namely, the one consisting of and their common neighbor at the interior face incident to ; this vertex is interior since has no chord and at least 4 vertices).
We say that a 2-leg-center dominates a 2-leg-center if vertex is strictly inside the cycle formed by some 2-leg with center-vertex . See also Figure 2(left). The dominance-relationship is acyclic since any 2-leg with center-vertex must enclose strictly fewer faces than the 2-leg . 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 , we have and so there exists at least one vertex between and on the outer-face. We say that a 2-leg is basic if the vertices all have degree 3, and complex otherwise. Note that if is basic, then form a fan and their common neighbor is .
Let be a minimal 2-leg center. We have two cases:
All 2-legs containing are basic.
Let be minimal and be maximal such that is adjacent to and . See also Figure 2(middle). Since is a 2-leg-center, we have . By case assumption the 2-leg is basic, so is a fan. We verify that is an internally 4-connected triangulated disk:
The outer-face of consists of the one of plus . By definition of a 2-center was not on the outer-face, so is a triangulated disk.
Since had no chord, the only possible chord of would be incident to vertex . But by choice of and the only neighbors of on the outer-face of are and . So has no chord.
Some 2-leg is complex.
We assume that has been chosen maximally, i.e., so that is either not a 2-leg or not complex. We claim that in this case is a suitable vertex set.
We first show that cannot be adjacent to . Assume for contradiction that it is, then is a triangle and hence a face. If there were some with and , then this would make a complex 2-leg, contradicting the choice of . So all of (if any) have degree 3, and they form a fan with common neighbor . In particular, edge exists, which means triangle is a face, forcing . But then is basic, not complex. This is a contradiction, so is not a neighbor of .
Let be the neighbors of in ccw order. See also Figure 2(right). None of can be on the outer-face of , else would have a chord. The outer-face of consists of , and so this is a simple cycle and is a triangulated disk. Further, we can show that it has no chord:
If a chord of connected two vertices in , then it would also be a chord in , which is excluded.
If a chord connected two non-consecutive vertices in , then in there would be an edge between two non-consecutive neighbors of , implying a triangle that is not a face.
If a chord connected some , , with some , , then would be a 2-leg in . By minimality of hence , but this contradicts that is not adjacent to .
If a chord connected some , , with some , or , then by it would have to cross or , contradicting planarity.
So is an internally 4-connected triangulated disk.
Observe that in both cases for some , and so does not contain or as desired. ∎
Let be a 4-connected planar triangulation. Then has a -canonical order.
We choose the vertex set in reverse order. Let be the outer-face and choose ; this satisfies all conditions since has at least 3 neighbors. (We do not at this point know the correct value of , but simply assign indices backwards and shift indices at the end so that the vertex sets are numbered .)
Observe that is an internally 4-connected triangulated disk, because the neighbors of form a simple cycle without chord (else there would be a separating triangle at ). Assume now some have been chosen already such that the remaining graph is an internally 4-connected triangulated disk with on the outer-face. If has at least 4 vertices, then apply Lemma 1 to find the next . Graph is again internally 4-connected, so we can continue choosing vertex sets until only 3 vertices, including and , are left. Since the graph is still internally 4-connected, these vertices must be a triangle, and hence a face of . So setting 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 is connected, it suffices to show that every vertex except has a neighbor in a later vertex set; the set of these edges then forms a spanning tree in . The argument for this is nearly the same as for -orderings. Clearly each of are adjacent to . For any vertex , vertex is not on the outer face of , and hence there must exist some minimal such that is on the outer face of , but not on the outer face of . Since faces are triangles, this implies that is adjacent to some vertex in . By the above hence is connected for any . ∎
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 -canonical ordering can be found in linear time; we omit the details.
In this section, we demonstrate two uses for the -canonical ordering in graph drawing. Both results proved here were known before, but in our opinion the -canonical ordering significantly simplifies the proof of these results.
4.1 Rectangular duals
A rectangular dual drawing (or RD-drawing for short) of a planar graph consists of a set of interior-disjoint rectangles assigned to the vertices of in such a way that the union of the rectangles forms a rectangle without holes, and the rectangles assigned to vertices and touch in a non-zero-length line segment if and only if is an edge. The following theorem has been proved repeatedly:
Let be a -connected triangulation, and let be an edge on the outer-face of . Then has a rectangular dual.
Previous proofs on this result usually used the -canonical ordering (or some equivalent characterization, such as regular edge labellings). We give here a different proof using the -canonical ordering.
Let the outer-face be , chosen such that . Find a -canonical ordering of . We now build the rectangular-dual drawing of by drawing for . By construction, is an edge on the outer-face of , and we can hence enumerate the outer-face of as with and . We maintain the invariant that in the RD-drawing of , the rectangles of all attach at the top side of the bounding box, in this order.
Such a drawing is easily created for , since is a triangle and so is a path , where is the third vertex of the interior face at . Now assume is drawn and consider adding either a singleton or a fan . Let and be the smallest and largest index such that and are adjacent to a vertex in .
Extend all rectangles of and upward by one unit. This leaves a “gap” where the rectangles of ended. There is at least one such rectangle since by properties of the -canonical ordering (else would not be 3-connected). If is a singleton , then we insert the rectangle for into this gap. If is a fan , then and so the gap consists exactly of the top of . Split this range into pieces and assign rectangles for 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. ∎
4.2 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 the rectangle defined by is empty, i.e., contains no other points of vertices of the graph. (It may contain parts of other edges.) Here, is the minimum axis-aligned rectangle that contains the points of and ; it degenerates into a line segment if or are on a horizontal or vertical line. The following result is known:
Theorem 3 ([Bbm99]).
Let be a -connected triangulation and let be one edge of the outer-face. Then has a (weak, closed) rectangle-of-influence drawing.
We re-prove this result using the -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 -canonical ordering.
Let the outer-face be , chosen such that . Find a -canonical ordering of . We now build the RI-drawing of by drawing for . By construction is an edge on the outer-face of , and we can hence enumerate the outer-face of as with and . We maintain the invariant that in the RI-drawing of
Such a drawing is easily created for , since is a triangle and so is a path , where is the third vertex of the interior face at . Now assume is drawn and consider adding either a singleton or a fan . Let be the smallest and be the largest index such that and are adjacent to a vertex in . By 3-connectivity of we have . If is a singleton , then define
See also Figure 4(middle). By adding this new point satisfies the invariant. All rectangles are empty for , because they do not intersect the drawing of except in rectangles and . So we have the desired RI-drawing.
If is a fan , then . For , define
See also Figure 4(right). By adding these new points satisfies the invariant. All rectangles are empty for , because they do not intersect the drawing of except in rectangles and . So we have the desired RI-drawing. ∎
We showed the existence of new canonical order for -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 -connected triangulation have an -canonical ordering for all and reasonable restrictions on vertex sets ? The missing case is a -canonical ordering for 5-connected triangulations.
The -canonical ordering definition naturally generalizes to planar graphs that are not necessarily triangulated. For the corresponding -orderings [Kan96] and -orderings [NRN97] it suffices to allow adding chains, i.e., induced paths. Are there -orderings, -orderings and -orderings for 4-connected/5-connected planar graphs with some simple subgraphs as vertex sets ? Likewise, exploration of -canonical orders for non-planar graphs for remains completely open.
- Some references instead define to be the subgraph induced by . This complicates stating some of the conditions.
- The proof is strongly inspired of the one for a -canonical order in 5-connected graphs [NN00]. Since we demand less on our -canonical order, we can simplify the exposition somewhat.
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