On r-Edge-Connected r-Regular Bricks and Braces and Inscribability

A classical result due to Steinitz states that a graph is isomorphic to the graph of some 3-dimensional polytope P if and only if it is planar and 3-connected. If a graph G is isomorphic to the graph of a 3-dimensional polytope inscribed in a sphere, it is said to be of inscribable type. The problem of determining which graphs are of inscribable type dates back to 1832 and was open until Rivin proved a characterization in terms of the existence of a strictly feasible solution to a system of linear equations and inequalities which we call sys(G), which, surprisingly, also appears in the context of the Traveling Salesman Problem. Using such a characterization, various classes of graphs of inscribable type can be described. Dillencourt and Smith gave a characterization of 3-connected 3-regular planar graphs that are of inscribable and a linear-time algorithm for recognizing such graphs. In this paper, their results are generalized to r-edge-connected r-regular graphs for odd r ≥ 3 in the context of the existence of strictly feasible solutions to sys(G). An answer to an open question raised by D. Eppstein concerning the inscribability of 4-regular graphs is also given.


Background and main results
Given a 3-dimensional polytope P , we define the graph of P , denoted by G(P ), to be the graph (V, E) where V is the set of extreme points of P and uv ∈ E if and only if u and v are adjacent in P .
Let G = (V, E) be an undirected simple graph. A classical result due to Steinitz [16] connecting graph theory to geometry is the following: Theorem 1. G is isomorphic to the graph of some 3-

dimensional polytope P in R 3 if and only if it is planar and 3-connected.
A 3-connected planar graph isomorphic to the graph of a 3-dimensional polytope inscribed in a sphere is said to be of inscribable type. Steinitz [17] gave examples of graphs that are not of inscribable type. The problem of determining which graphs are of inscribable type dates back to 1832 (see [15]) and was open until Hodgson et al. [10] announced the following in 1992:

Theorem 2. If G is 3-connected and planar, then G is of inscribable type if and only if there exists x ∈ R E
satisfying: x e < π ∀ e ∈ E. On r-Edge-Connected r-Regular Bricks and Braces and Inscribability inequalities strictly a strictly feasible solution. Using this terminology and with sys(G) denoting the system x(δ(S)) ≥ 2 ∀ S ⊂ V, 2 ≤ |S| ≤ |V | − 2, x ≥ 0, Theorem 2 can be rephrased as follows: Rivin gave two proofs of Theorem 2. One uses hyperbolic geometry [13]. The other is an elementary proof using mathematical optimization [14].
Incidentally, sys(G) also defines what is known as the subtour-elimination polytope of G in connection with the Traveling Salesman Problem. This remarkable, albeit accidental, connection between the subtour-elimination polytope and the century-old geometry problem provided a motivation for studying strictly feasible solutions to the system sys(G).
Using Theorem 3, one can show that various classes of graphs are of inscribable type. For instance, Dillencourt and Smith [3] showed that, among others, 4connected planar graphs and planar graphs obtained from 4-connected planar graphs by removing one vertex are of inscribable type. In [2], they gave necessary and sufficient conditions for a 3-connected 3-regular planar graph to be of inscribable type and a linear-time algorithm for recognizing such a graph. In particular, they showed the following:

then G is of inscribable type if and only if
G is more-than-1-tough or G is bipartite and has a 4connected dual.
In this paper, we extend the above result to the following: in the tight cut decomposition of a matching-covered graph, a procedure described in a landmark paper by Lovász [11] in the study of the matching lattice.) In addition to proving Theorems 5 and 6, we also give a characterization of when such graphs are bricks and braces; the characterization generalizes a notion introduced by Dillencourt and Smith for 3-connected 3-regular planar graphs.
Before we end this section, we remark that a simple rregular graph cannot be planar for r ≥ 6. In addition, a simple r-regular planar graph cannot be bipartite for r ≥ 4. Thus, specializing Theorem 6 to 5-edge-connected 5regular planar graphs gives: We mention in passing that 5-edge-connected 5regular simple bricks do exist and so they give a previously unrecognized class of graphs of inscribable type.
An example of such a graph can be found in [1].

Notation and definitions
Unless otherwise stated, graphs are assumed to be The following characterization is due to Tutte [18].

Theorem 8. G has a perfect matching if and only if for
An immediate consequence of the above theorem is the following:  [12], [5] and [11].) Using the result [5] that each tight cut in a brick is trivial, Lovász [11] showed:

Theorem 11. A matching-covered graph has no nontrivial tight cuts if and only if it is either a brick or a brace.
The set of solutions to sys(G) is denoted by SEP(G). We now show that dim(SEP(G)) ≥ dim(PM(G)).
Define the affine function f : Let M be any perfect matching of G.
We now prove the second part. Let C be a non-trivial , C is not a tight cut. The result now follows.
Proof of Theorem 6. Since x = 2 r e is a solution to sys(G) with x > 0, sys(G) has a strictly feasible solution if and only if G has no non-trivial constricted cut. By the second part of Proposition 13, G has no non-trivial constricted cut if and only if G has no non-trivial tight cut. The result now follows from Theorem 11 because G is matching-covered by Corollary 10 as 1 r ∈ PM(G).
Theorem 5 follows from Theorem 6 and the next two lemmas.

Lemma 14. If G is non-bipartite, then G is a brick if
and only if G is more-than-1-tough.

Lemma 15. If G is bipartite, then G is a brace if and
only if G has no non-trivial r-edge cuts.
Proof of Lemma 14. Observe that G is 3-connected and has at least four vertices. Therefore, it suffices to show that G is bicritical if and only if G is more-than- denote the vertex sets of the components of G − S.
Since G is r-edge-connected, |δ(S i )| ≥ r for i = 1, ..., k. Proof. Since G is r-regular, Let C be a non-trivial cut of G. Suppose that C is a tight cut. Then x(C) = 1 for all x ∈ PM(G). Since

A note on 4-regular graphs
So far, the results that have been discussed concern r-regular graphs where r is odd. When r is even, the situation is somewhat unclear and a characterization of all 3-connected 4-regular planar graphs of inscribable type using simple graph-theoretical terms is not yet known.
For example, with regards to 4-regular planar graphs, Eppstein [6] raised the following question: Is a morethan-1-tough 3-connected 4-regular planar graph of inscribable type? The answer is 'no' and the graph depicted in Figure 1 is more-than-1-tough but is not of For the next few lemmas, let (P ) denote the linear programming problem: max 0 subject to x ≥ 0 and let (D) denote the dual of (P ): Let sys ′ (G) denote the set of constraints in (D).
The next lemma gives a sufficient condition for a cut to be constricted and an edge to be useless. Proof. Since G is feasible, (P ) has an optimal solution.

Lemma 18. Let G be a feasible graph. If there exist
By strict complementarity for linear programming, there exist an optimal solutionȳ,z such that a cut A is con- Next, we obtain a refinement of Lemma 19 using the notion of uncrossing. Letȳ,z be integral and feasible for sys ′ (G). Let A(ȳ) denote the set {A ∈ C(G) :ȳ A > 0}.
Let δ(S) and δ(T ) be crossing cuts in A(ȳ). By uncrossing δ(S) and δ(T ), we mean applying the following modifications toȳ,z: such that Uncross A and B to obtain y ′ , z ′ . It is not difficult to see that y ′ , z ′ are still feasible for sys ′ (G) and ∑ joining a vertex in T and a vertex in U . Then, for anȳ x ∈ SEP(G), x(δ(S)) =x(δ(T )) +x(δ(U )) ≥ 2 + 2 = 4, contradicting that δ(S) is constricted.
The next result appears in Grünbaum [8].
From Lemma 24, one deduces that then G cannot be bipartite.
Proof of Theorem 17. Suppose that sys(G) has no strictly feasible solution. Since 1 2 e ∈ SEP(G), G is feasible and has no useless edge. Therefore, G must have a non-trivial constricted cut. By Lemma 21, there exist is non-crossing and non-empty. As A(ŷ) is non-crossing, To show that H is simple, we first prove the following: Claim. Let S ∈ S ′ . Then for every edge uv such that u ∈ S and v / ∈ S,ẑ u = 0 andẑ v > 0.
An immediate consequence of this claim is that if w is a neighbour of a pseudo-vertex in H, then w is not a pseudo-vertex andẑ w > 0.
To prove the claim, note that as G has no useless edge, by Lemma 18, for all pq ∈ E such that p, q ∈ S, we havê From this claim, one can see that δ(S 1 ) and δ(S 2 ) are disjoint for any distinct S 1 , S 2 ∈ S ′ . Thus contracting each element of S ′ does not create parallel edges. So H is simple. Also, the set of pseudo-vertices in H is independent.
To show that H is non-bipartite, first suppose that T = V . In this case, H is simple, connected, planar, and 4-regular and therefore is non-bipartite by Lemma 25.
Otherwise, H has exactly four vertices of degree three and no vertex of degree two. By Lemma 24, H has a triangle and therefore is non-bipartite.
Let v be a neighbour of a pseudo-vertex in H. By the claim above, v is not a pseudo-vertex andẑ v > 0. Let  Note that Theorem 17 does not hold if the condition to be planar is dropped. The graph depicted in Figure 3 is a 3-connected 4-regular graph G whose non-trivial 4-edge cuts are matchings of G but sys(G) has no strictly feasible solution. Note that the graph is not more-than-1tough and it has a non-trivial constricted cut. However, every non-trivial 4-edge cut of the graph is a matching.
One might ask what happens if we restrict our attention to more-than-1-tough 4-regular graphs. We do not know the answer and so we have the following problem: Problem 26. Let G be a more-than-1-tough 4-regular graph. If every non-trivial 4-edge cut of G is a matching, must sys(G) have a strictly feasible solution?