Flip-distance between \alpha-orientations of graphs embedded on plane and sphere

TL;DR

This paper provides an explicit formula for the minimum number of flip operations needed to transform one lpha-orientation into another in graphs embedded on the plane or sphere, extending previous lattice structure results.

Contribution

It introduces a formula for the flip-distance between lpha-orientations on planar and spherical graphs, advancing understanding of their combinatorial structure.

Findings

Derived an explicit flip-distance formula for lpha-orientations.

Extended lattice structure analysis to include flip distances.

Applicable to graphs embedded on both plane and sphere.

Abstract

Felsner introduced a cycle reversal, namely the `flip' reversal, for \alpha-orientations (i.e., each vertex admits a prescribed out-degree) of a graph G embedded on the plane and further proved that the set of all the \alpha-orientations of G carries a distributive lattice with respect to the flip reversals. In this paper, we give an explicit formula for the minimum number of flips needed to transform one \alpha-orientation into another for graphs embedded on the plane or sphere, respectively.

Equations

$$\left|{\mathcal{F}}_{\mathrm{int}}(C^{-})\setminus \bigcup _{i=1}^t{\mathcal{F}}_{\mathrm{int}}(C_i^{+})\right|$$

$$P=P_1(u\to v)\cup C_1^{+}(v\to v^{\prime })\cup P_2(v^{\prime }\to u^{\prime }),$$

$$C_1=P(u\to v_1)\cup C_1^{+}(v_1\to v_1^{\prime })\cup P(v_1^{\prime }\to v_2)\cup C_2^{+}(v_2\to v_2^{\prime })\cup \cdots$$

$$\cup C_k^{+}(v_k\to v_k^{\prime })\cup P(v_k^{\prime }\to u^{\prime })\cup C^{-}(u^{\prime }\to u).$$

$$\left|{\mathcal{F}}_{\mathrm{int}}(C_1)\setminus \bigcup _{{\mathcal{F}}_{\mathrm{int}}(C_i^{+})\subset {\mathcal{F}}_{\mathrm{int}}(C_1)}{\mathcal{F}}_{\mathrm{int}}(C_i^{+})\right|$$

$$C_2=P^{-1}(u^{\prime }\to v_k^{\prime })\cup C_k^{+}(v_k^{\prime }\to v_k)\cup P^{-1}(v_k\to v_{k-1}^{\prime })\cup C_{k-1}^{+}(v_{k-1}^{\prime }\to v_{k-1})\cup \cdots$$

$$\cup C_1^{+}(v_1^{\prime }\to v_1)\cup P^{-1}(v_1\to u)\cup C^{-}(u\to u^{\prime })$$

$$\left|{\mathcal{F}}_{\mathrm{int}}(C_2)\setminus \bigcup _{{\mathcal{F}}_{\mathrm{int}}(C_i^{+})\subset {\mathcal{F}}_{\mathrm{int}}(C_2)}{\mathcal{F}}_{\mathrm{int}}(C_i^{+})\right|$$

$${\mathcal{F}}_{\mathrm{int}}(C_1)\cup {\mathcal{F}}_{\mathrm{int}}(C_2)={\mathcal{F}}_{\mathrm{int}}(C^{-})\setminus \bigcup _{i=1}^k{\mathcal{F}}_{\mathrm{int}}(C_i^{+}).$$

$$\{C_i^{+}:{\mathcal{F}}_{\mathrm{int}}(C_i^{+})\subset {\mathcal{F}}_{\mathrm{int}}(C_1)\}\cup \{C_i^{+}:{\mathcal{F}}_{\mathrm{int}}(C_i^{+})\subset {\mathcal{F}}_{\mathrm{int}}(C_2)\}\cup \bigcup _{i=1}^k\{C_i^{+}\}={\mathcal{C}}^{+}.$$

$$z_F(f_2)=\{\begin{array}{ll}{z}_F(f_1)+1,& \text{if e∈F and f2 is left of e},\\ z_F(f_1)-1,& \text{if e∈F and f1 is left of e},\\ z_F(f_1),& \text{otherwise}.\end{array}$$

$$z_F(f)=c^{-}-c^{+},$$

$$z_{D^{\prime }-D}(f)\ge 0.$$

$$d(D^{\prime },D)=\sum _{f\in \mathcal{F}(G)}{z}_{D^{\prime }-D}(f).$$

$$z_{(D^{\prime }-D{)}^{\ast }}(f)=\{\begin{array}{ll}{z}_{(D^{\prime }-D)}(f)-1,& \text{if f∈Fint(C−)∖⋃i=1tFint(Ci+)},\\ z_{(D^{\prime }-D)}(f),& \text{otherwise}.\end{array}$$

$$f\in {\mathcal{F}}_{\mathrm{int}}(C^{-})\setminus \bigcup _{i=1}^t{\mathcal{F}}_{\mathrm{int}}(C_i^{+})$$

$$d(D^{\prime },D)=|F_1\cup F_2|=\sum _{f\in \mathcal{F}(G)}t(f)=\sum _{f\in \mathcal{F}(G)}{z}_{D^{\prime }-D}(f),$$

$$d(D^{\prime },D)=\sum _{i=1}^s|{\mathcal{F}}_{\mathrm{int}}(C_i^{-})|-\sum _{j=1}^t|{\mathcal{F}}_{\mathrm{int}}(C_j^{+})|.$$

$$z_{D^{\prime }-D}(f,g):\mathcal{F}(G)\to \mathbf{Z}$$

$$z_{D^{\prime }-D}(f,g_2)=\{\begin{array}{ll}{z}_{D^{\prime }-D}(f,g_1)+1,& \text{if e∈D′−D and g2 is left of e},\\ z_{D^{\prime }-D}(f,g_1)-1,& \text{if e∈D′−D and g1 is left of e},\\ z_{D^{\prime }-D}(f,g_1),& \text{otherwise}.\end{array}$$

$$d(D^{\prime },D)=\sum _{g\in \mathcal{F}(G)}(z_{D^{\prime }-D}(f,g)-z_{\min }(f)),$$

$$z_{\min }(f)=\min \{z_{D^{\prime }-D}(f,g):g\in \mathcal{F}(G)\}).$$

$$z^{\prime }(g)-z^{\prime }(g^{\prime })=z_{D^{\prime }-D}(f,g)-z_{D^{\prime }-D}(f,g^{\prime }).$$

$$g=g_1,g_2,\cdots ,g_p=g^{\prime },$$

$$z_{D^{\prime }-D}(f,g^{\prime })=z_{\min }(f).$$

$$z^{\prime }(g)-z_{D^{\prime }-D}(f,g)=z^{\prime }(g^{\prime })-z_{D^{\prime }-D}(f,g^{\prime })=z^{\prime }(g^{\prime })-z_{\min }(f).$$

$$d(D^{\prime },D)=q=\sum _{g\in \mathcal{F}(G)}{z}^{\prime }(g)=\sum _{g\in \mathcal{F}(G)}(z_{D^{\prime }-D}(f,g)+z^{\prime }(g^{\prime })-z_{\min }(f))$$

$$\ge \sum _{g\in \mathcal{F}(G)}(z_{D^{\prime }-D}(f,g)-z_{\min }(f)),$$

$$g\in {\mathcal{F}}_{\mathrm{inner}}(G_{C_0}^{+})\setminus \bigcup _{i=1}^t{\mathcal{F}}_{\mathrm{int}}(C_i),$$

$$z_{D^{\prime }-D}(f,g)=z_{D^{\prime }-D}(f,g_0)=z_{\min }(f).$$

Full text

Keywords: $\alpha$-orientation; flip distance; plane graph; sphere graph

1 Introduction

Research in graph orientation has a long history that reveals many interesting structural insights and applications. A classical example would be the one given by Robbins in 1939, which states that an undirected graph has a strongly connected orientation if and only if it is 2-edge connected. This result was then generalized by Nash-Williams to strongly $k$-edge connected orientations for any positive $k$ [18]. In the study of graph orientation, a particular concern is to orient a graph with certain degree-constraints on its vertices. Frank [7] established a characterization for the existence of those orientations in which the in-degree of each vertex has to lie within certain bounds. In [12], Hakimi gave a characterization for a graph to have an orientation such that each vertex has a prescribed out-degree. Such orientation is later called the $\alpha$-orientation [5].

Graph orientation has many interesting connections with certain combinatorial structures in graphs, such as the spanning trees [5], bipartite perfect matchings (or more generally, bipartite $f$-factors) [14, 16, 19], Schnyder woods [8], bipolar orientations [9] and 2-orientations of quadrangulations, primal-dual orientations [4], transversal structures [10] and $c$-orientations of the dual of plane graph [15, 19]. Remarkably, all these structures can be encoded as the $\alpha$-orientations [6], which has extensive applications, e.g., in drawing algorithms [1, 2, 10], and enumeration and random sampling of graphs [3, 11].

To deal with the relation among orientations, cycle reversal has been shown as a powerful method since it preserves the out-degree of each vertex and the connectivity of the orientations. Various types of cycle reversals were introduced subject to certain problem-specific requirements. An earlier example is the cycle transformation introduced by Nash-Williams [18] which says that any two $k$-connected orientations of a $2k$-edge connected graph can be transformed from each other by a sequence of cycle transformations or path transformations.

Cycle reversal for the orientations of plane graphs (graphs embedded on the plane) received particular attention. For example, Nakamoto [17] considered the 3-cycle reversal to deal with the $(*)$-orientations in plane triangulation where each vertex on the outer facial cycle has out-degree 1 while each of the other vertices has out-degree 3. In [20], Zhang et al. introduced the Z-transformation to study the connection among perfect matchings of hexagonal systems and later was extended to general plane bipartite graphs [21].

For orientations of general plane graphs, a natural considering of cycle reversal is to reverse a directed facial cycle. However, an orientation of a plane graph does not always have such a directed facial cycle, even if it has ‘many directed cycles’. In [5], Felsner introduced a type of cycle reversal, namely the ‘flip’, defined on the so-called essential cycles and proved that, any $\alpha$-orientation of a plane graph $G$ can transform into a particular $\alpha$-orientation by a flip sequence and further proved that the set of all $\alpha$-orientations of $G$ carries a distributive lattice with respect to the flip reversals. In a strongly connected $\alpha$-orientation of a plane graph, it is known that an essential cycle is exactly an inner facial cycle. In this sense, the notion ‘essential cycle’ is a very nice generalization of facial cycle, which has been widely applied in the study of $\alpha$-orientations of plane graphs.

In this paper, we give a necessary-sufficient condition for that an $\alpha$-orientation of a plane graph can be transform into another by a flip sequence. In contrast to a plane graph, we will see in the last section that any two $\alpha$-orientations of a sphere graph (a graph embedded on the sphere) can always be transformed from each other by a flip sequence. Further, we give an explicit formula of the ‘flip-distance’ between two $\alpha$-orientations, that is, the minimum number of flips needed to transform one $\alpha$-orientation into another, for plane graphs and sphere graphs, respectively. In our study, the ‘standard cycle system’ introduced in the following section and the idea of ‘length function’ [13] and ‘$\alpha$-potential’ [5] defined on the faces of the embedded graph play important roles.

2 Preliminaries

For a graph $G$ we denote by $V(G)$ and $E(G)$ the vertex set and edge set of $G$, respectively. An orientation of $G$ is an assignment of a direction to each edge of $G$. A plane graph and sphere graph are an embedding of a planar graph on the plane and sphere, respectively. Given a plane graph $G$ with $n$ vertices $v_{1},v_{2},\cdots,v_{n}$ and an out-degree function $\alpha=(\alpha(v_{1}),\alpha(v_{2}),\cdots,\alpha(v_{n}))$ of $G$, an orientation $D$ of $G$ is called an $\alpha$-orientation if $d^{+}_{D}(v_{i})=\alpha(v_{i})$ for all $v_{i}\in V(G)$, where $d^{+}_{D}(v_{i})$ is the out-degree of $v_{i}$ in $D$. We call an out-degree function $\alpha$ feasible for a graph $G$ if $\alpha$-orientation of $G$ exists. The question whether an out-degree function $\alpha$ is feasible for $G$ can be translated to the construction of a maximal flow in a graph associated with $G$, and therefore can be solved in polynomial time [5].

An edge in a graph $G$ is called $\alpha$-rigid if it has the same direction in every $\alpha$-orientation of $G$. For a cycle $C$ of a plane graph $G$, the interior cut of $C$ is the edge cut consisting of all the edges connecting $C$ to an interior vertex of $C$. A simple cycle $C$ of a plane graph $G$ is called essential [5] (with respect to $\alpha$) if $C$ is chord-free, all the edges in the interior cut of $C$ are rigid and there exists an $\alpha$-orientation such that $C$ is directed.

A directed cycle $C$ of an $\alpha$-orientation of a plane graph is called counterclockwise (resp., clockwise), or ccw (resp., cw) for simplicity, if the interior region of $C$ is to the left (resp., right) of $C$. A flip taking on an essential ccw cycle $C$ is the reorientation of $C$ from ccw to cw [5]. The flip reversal induces a partial ordering relation ‘$\preccurlyeq$’ on the set of all $\alpha$-orientations of a plane graph, that is, two $\alpha$-orientations $D$ and $D^{\prime}$ have the relation $D\preccurlyeq D^{\prime}$ if $D$ can be transformed from $D^{\prime}$ by a sequence of flips. We denote by $d(D^{\prime},D)$ the flip-distance from $D^{\prime}$ to $D$ if $D\preccurlyeq D^{\prime}$.

By the definition of rigid edge, we can see that each component obtained from an $\alpha$-orientation $D$ by deleting all the rigid edges is strongly connected [15]. To simplify our discussion, in the following we restrict our attention only to strongly connected orientations and therefore, the underlying graph $G$ is 2-edge connected. Further, if $G$ is not 2-connected then a strongly connected $\alpha$-orientation $D$ of $G$ is the edge-disjoint union of the restriction of $D$ on the 2-connected components of $G$. For this reason, we only consider the case that $G$ is 2-connected and, therefore, any inner facial cycle $f$ (boundary of an inner face) in $G$ is simple, that is, each vertex on $f$ appears only once.

As mentioned earlier, in a strongly connected $\alpha$-orientation of a plane graph, since every edge is not rigid, an essential cycle now is exactly a facial cycle. We notice that this is also the case for a sphere graph. Therefore, the notion ‘flip’ for the ccw facial cycles and the ‘flip-distance’ for the $\alpha$-orientations of a sphere graph can be naturally defined. Further, for a sphere graph $G$, only in the very special case when $G$ is a simple cycle $C$, $C$ is the facial cycle of two faces of $G$ and thus, in any orientation of $G$, $C$ is ccw with respect to one face if and only if $C$ is cw with respect to the other.

For a plane or sphere graph $G$, we denote by ${\cal F}(G)$ the set of all the faces of $G$. In particular, if $G$ is a plane graph then we denote by $f_{\rm out}(G)$ and ${\cal F}_{\rm inner}(G)$ the outer face and the set of all the inner faces of $G$, respectively. For a cycle $C$ in a plane graph $G$, we denote by ${\cal F}_{\rm int}(C)$ the set of the faces in the interior region of $C$. For two edge-disjoint cycles $C_{1}$ and $C_{2}$ of a plane graph, if ${\cal F}_{\rm int}(C_{2})\subset{\cal F}_{\rm int}(C_{1})$ then, except some possible vertices in common with $C_{1}$, the cycle $C_{2}$ lies in the interior region of $C_{1}$. In this sense, we also say that $C_{2}$ is contained in $C_{1}$, or conversely, $C_{1}$ contains $C_{2}$.

The following result will be used in our forthcoming argument.

Lemma 2.1**.**

[5]** If $C$ is a ccw cycle in a strongly connected $\alpha$-orientation $D$ of a 2-connected plane graph, then there is a flip sequence $F$ consisting of $|{\cal F}_{\rm int}(C)|$ flips which reverses $C$ from ccw into cw and keep the orientations of all other edges of $D$ invariant. More specifically, the number of the flips in $F$ taking on each inner face $f$ equals 1 if $f\in{\cal F}_{\rm int}(C)$, and equals 0 if $f\notin{\cal F}_{\rm int}(C)$.

A set ${\cal C}$ of edge disjoint directed cycles in a directed plane graph or sphere graph is called a standard cycle system if any two cycles in ${\cal C}$ are pairwise uncrossed. In the case of plane graph, we see that any two cycles $C_{1}$ and $C_{2}$ in a standard cycle system satisfy either ${\cal F}_{\rm int}(C_{1})\cap{\cal F}_{\rm int}(C_{2})=\emptyset$, or ${\cal F}_{\rm int}(C_{1})\subset{\cal F}_{\rm int}(C_{2})$, or ${\cal F}_{\rm int}(C_{2})\subset{\cal F}_{\rm int}(C_{1})$. A directed graph $F$ (not necessarily connected) is called oriented Eulerian if each component of $F$ is directed Eulerian, that is, the out-degree of each vertex equals its in-degree. The following lemma might be a known result but we give its proof for the self-completeness.

Lemma 2.2**.**

Any oriented Eulerian plane graph or sphere graph $F$ can be partitioned into the union of pairwise edge disjoint and uncrossed directed cycles, that is, a standard cycle system.

Proof.

We apply induction on $|{\cal F}(F)|$, that is, the number of the faces in $F$.

Let $f$ be a directed facial cycle of $F$ (the existence of $f$ is obvious since $F$ is oriented Eulerian). Then $F\setminus E(f)$ is still an oriented Eulerian graph, where $F\setminus E(f)$ is the subgraph of $F$ obtained from $F$ by removing the edges on $f$. Moreover, $F\setminus E(f)$ has less faces than $F$ has. So by the induction hypothesis, $F\setminus E(f)$ can be partitioned into a standard cycle system ${\cal C}$. Notice that the cycles in ${\cal C}$ and the facial cycle $f$ are edge disjoint and uncrossed. This means that ${\cal C}\cup\{f\}$ is a standard cycle system of $F$, which completes our proof. ∎

3 Graphs embedded on the plane

Lemma 3.1**.**

Let $D$ be a strongly connected $\alpha$-orientation of a 2-connected plane graph $G$. Let $C^{-}$ be a simple ccw cycle in $D$ and let ${\cal C}^{+}=\{C_{1}^{+},C_{2}^{+},\cdots,$ $C^{+}_{t}\}$ be the set of edge disjoint, uncrossed and pairwise exclusive simple cw cycles contained in $C^{-}$. Then there is a flip sequence $F$ consisting of

[TABLE]

flips which reverses all cycles $C^{-},C_{1}^{+},C_{2}^{+},\cdots,C^{+}_{t}$ and keep the orientations of all other edges of $D$ invariant. More specifically, for any face $f$, the number of flips in $F$ taking on $f$ equals 1 if $f\in{\cal F}_{\rm int}(C^{-})\setminus\bigcup_{i=1}^{t}{\cal F}_{\rm int}(C^{+}_{i})$, and equals 0 otherwise.

Proof.

We apply induction on the number $t$. The assertion follows directly by Lemma 2.1 if $t=0$.

Since $D$ is strongly connected, there is a directed path $P_{1}$ from a vertex $u$ on $C^{-}$ to a vertex $v$ on $C^{+}_{1}$ (in the degenerated case when $C^{-}$ and $C^{+}_{1}$ has a common vertex, we may have $u=v$) and a directed path $P_{2}$ from a vertex $v^{\prime}$ on $C^{+}_{1}$ to a vertex $u^{\prime}$ on $C^{-}$. Moreover, we may assume that $P_{1}$ and $P_{2}$ are shortest. This means that, $u$ (resp., $v$) is the only common vertex of $P_{1}$ and $C^{-}$ (resp., $C^{+}_{1}$) while $u^{\prime}$ (resp., $v^{\prime}$) is the only common vertex of $P_{2}$ and $C^{-}$ (resp., $C^{+}_{1}$). Let

[TABLE]

where for a directed path or cycle $W$ and two vertices $u$ and $v$ on $W$, $W(u\rightarrow v)$ is the section of $W$ from $u$ to $v$. Thus, $P$ is a directed path that visits $C^{+}_{1}$ exactly once.

We notice that $P$ may visit more cycles in ${\cal C}^{+}$ other than $C^{+}_{1}$, say $C^{+}_{2},\cdots,$ $C^{+}_{k}$ and, without loss of generality, we assume that $P$ successively visits $C^{+}_{1},C^{+}_{2},$ $\cdots,C^{+}_{k}$, see Figure 1(a). Moreover, along with the directed path $P$, we may assume that $P$ visits $C^{+}_{i}$ exactly once and that $v_{i}$ (resp., $v^{\prime}_{i}$) is the first (resp., last) vertex on $C^{+}_{i}$ for each $i\in\{1,2,\cdots,k\}$. On the other hand, since $P$ is shortest, $u$ and $u^{\prime}$ are the only common vertices of $P$ and $C^{-}$.

Figure 1.

Consider the directed cycle

[TABLE]

[TABLE]

Clearly $C_{1}$ is a ccw cycle (possibly not simple) and contains less number of cw cycles in ${\cal C}^{+}$ than $C^{-}$ does. Moreover, since $G$ is 2-connected and $D$ is strongly connected, we can see that the subgraph of $G$ induced by the vertices on $C_{1}$ and in the interior region of $C_{1}$ is still 2-connected and the sub-orientation of $D$ restricted on the sugraph is still strongly connected. Thus, if $C_{1}$ is simple then by the induction hypothesis, $C_{1}$ and those cw cycles $C^{+}_{i}$ contained in $C_{1}$ can be reversed by a flip sequence $F_{1}$ consisting of

[TABLE]

flips. If $C_{1}$ is not simple, i.e., $u=u^{\prime}$, then we can split $u=u^{\prime}$ into two different vertices $u$ and $u^{\prime}$ so that $C_{1}$ become to be a simple cycle $C^{*}_{1}$, see Figure 1(b). Notice that the ‘splitting’ does not effect any flips since a flip only involves the edges of a facial cycle. Moreover, ${\cal F}_{\rm int}(C^{*}_{1})={\cal F}_{\rm int}(C_{1})$. The remaining discussion is analogous.

After $F_{1}$ being taken, the cycle

[TABLE]

[TABLE]

is ccw, where $P^{-1}$ is the reversal of $P$. Again by the induction hypothesis, $C_{2}$ and those $C^{+}_{i}$ contained in $C_{2}$ can be reversed by a flip sequence $F_{2}$ consisting of

[TABLE]

flips. By the construction of $C_{1}$ and $C_{2}$ we have

[TABLE]

Moreover,

[TABLE]

Hence, (1) follows directly from (2) and (3). This completes the proof. ∎

Let $f_{1}$ and $f_{2}$ be two adjacent faces and let $e$ be a common edge of $f_{1}$ and $f_{2}$. Along with the direction of $e$, if $f_{1}$ (resp., $f_{2}$) lies to the left side of $e$ then we say that $f_{1}$ (resp., $f_{2}$) is left of $e$.

Following the idea of ‘length function’ introduced in [13] and ‘$\alpha$-potential’ introduced in [5], we give the following definition.

Definition 3.1 For an oriented Eulerian subgraph $F$ of an $\alpha$-orientation $D$, define the $F$-potential $z_{F}(f)$ of $D$ as a function which assigns an integer to each face $f$ of $D$ according to the following rule:

1. $z_{F}(f_{\rm out})=0$;

2. if $f_{1}$ and $f_{2}$ are two faces sharing a common edge $e$ then

[TABLE]

Let ${\cal C}$ be a standard cycle system of $F$. By the definition of $z_{F}(f)$, for any face $f$, one can see that the value of $z_{F}(f)$ is determined by those cycles in ${\cal C}$ that contain $f$ as an interior face. More specifically,

[TABLE]

where $c^{-}$ and $c^{+}$ are the numbers of ccw and cw cycles in ${\cal C}$ which contain $f$ as an interior face, respectively, as illustrated in Figure 2. We notice that the value of $c^{-}-c^{+}$ is a constant for any standard cycle system ${\cal C}$ of $F$. This means that $z_{F}(f)$ is determined uniquely by $F$ and therefore, is well defined.

Figure 2. The directed cycles in ${\cal C}$ are indicated in thick circles and the outer facial cycle is indicated in the thin circle.

For two $\alpha$-orientations $D$ and $D^{\prime}$, let $D^{\prime}-D$ be obtained from $D^{\prime}$ by removing those edges which has the same direction with $D$. We note that $D^{\prime}-D$ is oriented Eulerian.

Theorem 3.2**.**

Let $D$ and $D^{\prime}$ be two strongly connected $\alpha$-orientations of a 2-connected plane graph $G$. Then $D\preccurlyeq D^{\prime}$ if and only if, for any face $f\in{\cal F}(G)$,

[TABLE]

Moreover, if $D\preccurlyeq D^{\prime}$ then

[TABLE]

Proof.

Assume firstly that $D\preccurlyeq D^{\prime}$. Let $F=\{f_{1},f_{2},\cdots,f_{q}\}$ be a flip sequence that transforms $D^{\prime}$ into $D$.

For any face $f\in{\cal F}(G)$, consider the times $t(f)$ of flips in $F$ taking on $f$. Since the outer face $f_{\rm out}(G)$ is not involved in any flip, we have $t(f)=0$ if $f=f_{\rm out}(G)$. Now assume that $f^{\prime}$ is a face which shares a common edge $e$ with $f$. If $e\in D^{\prime}-D$, then the orientation of $e$ changes after $F$ being taken. Moreover, a flip to change the orientation of $e$ must be taken on the left face of $e$. This implies that $t(f)=t(f^{\prime})+1$ if $f$ is left of $e$ or $t(f)=t(f^{\prime})-1$ if $f^{\prime}$ is left of $e$. If $e\notin D^{\prime}-D$, then the orientation of $e$ does not change after $F$ being taken. This means that, if a flip in $F$ takes on $f$ then there is another flip that takes on $f^{\prime}$ to keep the orientation of $e$ invariant and, hence, $t(f)=t(f^{\prime})$. As a result, we have $t(f)=z_{D^{\prime}-D}(f)$. Notice that $t(f)\geq 0$ for any $f\in{\cal F}(G)$. The necessity follows.

Conversely, assume that (5) holds. Let ${\cal C}$ be a standard cycle system of $D^{\prime}-D$. If all the cycles in ${\cal C}$ are ccw then $D\preccurlyeq D^{\prime}$ follows directly by Lemma 2.1. Now let $C^{+}_{1}$ be a maximal cw cycle in ${\cal C}$, that is, there is no other cw cycle containing $C^{+}_{1}$ in ${\cal C}$. Then by (5), there is a ccw cycle $C^{-}\in{\cal C}$ satisfying ${\cal F}_{\rm int}(C^{+}_{1})\subset{\cal F}_{\rm int}(C^{-})$ and, we choose $C^{-}$ to be minimal. Notice that $C^{-}$ may contain one more such cw cycles. We denote by $C^{+}_{1},C^{+}_{2},\cdots,C^{+}_{t}$ all the maximal cw cycles of ${\cal C}$ which are contained in $C^{-}$.

Further, we notice that $D^{\prime}$ is strongly connected and $C^{-},C^{+}_{1},C^{+}_{2},\cdots,C^{+}_{t}$ are in $D^{\prime}$. So by Lemma 3.1, there is a flip sequence $F_{1}$ that reverses $C^{-}$ and $C^{+}_{i},i=1,2,\cdots,t$ and keep the orientations of all other edges of $D^{\prime}$ invariant. Let $(D^{\prime}-D)^{*}={\cal C}\setminus\{C^{-},C^{+}_{1},C^{+}_{2},\cdots,C^{+}_{t}\}$. Then by the choice of $C^{-},C^{+}_{1},$ $C^{+}_{2},\cdots,C^{+}_{t}$, for any face $f$, we have

[TABLE]

Moreover, by the maximality of $C^{+}_{1},C^{+}_{2},\cdots,C^{+}_{t}$, if

[TABLE]

then a cycle in ${\cal C}$ that contains $f$ as an interior face must be ccw. So by the definition of $z_{(D^{\prime}-D)}(f)$, we have $z_{(D^{\prime}-D)}(f)\geq 1$.

The above discussion implies that $z_{(D^{\prime}-D)^{*}}(f)\geq 0$ for any $f\in{\cal F}_{\rm int}(C^{-})$. So by a simple induction on the number of the cycles in ${\cal C}$, there is a flip sequence $F_{2}$ that reverses all cycles in $(D^{\prime}-D)^{*}$.

Thus, $F_{1}\cup F_{2}$ reverse all the cycles in ${\cal C}$, that is, transform $D^{\prime}$ into $D$. The sufficiency follows.

Finally, notice that $|F_{1}\cup F_{2}|$ equals the sum of the times $t(f)$ of flips in $F_{1}\cup F_{2}$ taking on each face $f\in{\cal F}(G)$. So by the proof of the necessity,

[TABLE]

which completes our proof. ∎

The following is an equivalent but more intuitive representation of (6).

Corollary 3.3**.**

Let $D$ and $D^{\prime}$ be two strongly connected $\alpha$-orientations of a 2-connected plane graph $G$ with $D\preccurlyeq D^{\prime}$, and let $C_{1}^{-},C_{2}^{-},\cdots,C_{s}^{-}$ and $C_{1}^{+},C_{2}^{+},\cdots,C_{t}^{+}$ be all the ccw and cw cycles in a standard cycle system of $D^{\prime}-D$, respectively. Then

[TABLE]

4 Graphs embedded on the sphere

From the graph embedding point of view, a graph embedded on the sphere is essentially the same as embedded on the plane. However, since a flip of an $\alpha$-orientation of a graph embedded on the sphere may be taken on any face of the graph while a flip for a graph embedded on the plane takes only on its inner face, the flip-distances for these two types of graph embedding are different. More specifically, we will see that, in contrast to plane graphs, any two $\alpha$-orientations of a sphere graph can always be transformed from each other by a flip sequence.

Let $D$ and $D^{\prime}$ be two $\alpha$-orientations of a sphere graph $G$. Choose an arbitrary face $f\in{\cal F}(G)$, similar to the $(D^{\prime}-D)$-potential $z_{D^{\prime}-D}(f)$ for plane graph, we define

[TABLE]

to be the function which assigns an integer to each face $g$ of $G$ according to the following rule, see Figure 3:

1. $z_{D^{\prime}-D}(f,f)=0$;

2. if $g_{1}$ and $g_{2}$ are two faces sharing a common edge $e$ then

[TABLE]

Figure 3.

For any $g\in{\cal F}(G)$, similar to the $F$-potential on plane graph, we can see that $z_{D^{\prime}-D}(f,g)$ is well defined.

Theorem 4.1**.**

Let $D$ and $D^{\prime}$ be two strongly connected $\alpha$-orientations of a 2-connected sphere graph $G$ and let $f$ be an arbitrary face of $G$ Then

[TABLE]

where

[TABLE]

Proof.

Let $F$ be a flip sequence with minimum number of flips, say $q$ flips, that transform $D^{\prime}$ into $D$ For any $g\in{\cal F}(G)$, let $z^{\prime}(g)$ be the number of flips in $F$ that takes on $g$ For any two adjacent faces $g$ and $g^{\prime}$, from the definition of $z_{D^{\prime}-D}(f,g)$ we can easily see that

[TABLE]

That is, $z_{D^{\prime}-D}(f,g_{1})-z_{D^{\prime}-D}(f,g_{2})$ depends only on $D$ and $D^{\prime}$, and does not depend on the choice of $f$ In general, for any two faces $g$ and $g^{\prime}$ (not necessarily adjacent), we can always find a face sequence

[TABLE]

such that $g_{i}$ and $g_{i+1}$ are adjacent for each $i\in\{1,2,\cdots,p-1\}$. This means that (6) holds for any two faces $g$ and $g^{\prime}$ of $G$. In particular, let $g^{\prime}$ be a face such that

[TABLE]

Then we have

[TABLE]

Therefore, the distance from $D^{\prime}$ to $D$ satisfies

[TABLE]

[TABLE]

where the last inequality holds because $z^{\prime}(g)\geq 0$ for any face $g\in{\cal F}(G)$

We now need only to find a flip sequence consisting of $\sum\limits_{g\in{\cal F}(G)}(z_{D^{\prime}-D}(f,g)-z_{\min}(f))$ flips which transform $D^{\prime}$ into $D$

Let ${\cal C}$ be a standard cycle system of $D^{\prime}-D$

For a directed cycle $C\in{\cal C}$, we can see that $C$ divides $G$ into two semi-sphere parts. Along with the orientation of $C$, let $G^{+}_{C}$ denote the subgraph of $G$ which lies in the right semi-sphere, including $C$ itself Similarly, let $G^{-}_{C}$ denote the subgraph of $G$ which lies in the left semi-sphere including $C$ itself In this way, $G^{+}_{C}$ and $G^{-}_{C}$ could be considered as two plane graphs. Moreover, $C$ is cw in $G^{+}_{C}$ and ccw in $G^{-}_{C}$ For convenience, we also use $G^{+}_{C}$ and $G^{-}_{C}$ to denote the ‘sub-orientation’ of $D^{\prime}$ restricted on $G^{+}_{C}$ and $G^{-}_{C}$, respectively.

Choose two adjacent faces $g_{0},g_{1}\in{\cal F}(G)$ such that $z_{D^{\prime}-D}(f,g_{0})=z_{\min}(f)$ and $z_{D^{\prime}-D}(f,g_{1})>z_{\min}(f)$. We note that, by the definition of $z_{D^{\prime}-D}(f,g)$, such pair of $g_{0}$ and $g_{1}$ exists since ${\cal C}$ is not empty. Let $e$ be a common edge shared by $g_{0}$ and $g_{1}$. Then, $e$ must be on a cycle in ${\cal C}$, say $C_{0}$ as shown in Figure 4(a). We draw $G^{-}_{C_{0}}$ and $G^{+}_{C_{0}}$ on the plane as shown in Figure 4(b) and (c), respectively.

In the plane graph $G^{+}_{C_{0}}$, with no loss of generality, let $C_{1},C_{2},$ $\cdots,C_{t}$ be all the maximal cycles of the cycle system ${\cal C}\setminus\{C_{0}\}$, that is, each $C_{i}$ is not contained in any other cycles of ${\cal C}\setminus\{C_{0}\}$. Since ${\cal C}$ is a standard cycle system of $D^{\prime}-D$, the interior region $B$ bounded by $C_{0},C_{1},C_{2},\cdots,C_{t}$ does not

Figure 4. The interior region $B$ is indicated in grey.

contain any cycle in ${\cal C}$. This means that, for any face

[TABLE]

we have

[TABLE]

Thus, again by the definition of $z_{D^{\prime}-D}(f,g)$, the directed cycles $C_{1},C_{2},\cdots,$ $C_{t}$ are all ccw in the plane graph $G^{+}_{C_{0}}$. Moreover, the plane graphs $G^{-}_{C_{i}},i=0,1,2,\cdots,t$, have the following properties:

1. a facial cycle in $G^{-}_{C_{i}}$ is ccw if and only if this facial cycle is ccw in $G$;

2. $G^{-}_{C_{0}}\cup G^{-}_{C_{1}}\cup\cdots\cup G^{-}_{C_{t}}$ contains all the cycles of ${\cal C}$;

3. $G^{-}_{C_{i}}$ is strongly connected.

For $i\in\{0,1,2,\cdots,t\}$, let $D_{i}$ and $D^{\prime}_{i}$ be the orientations $D$ and $D^{\prime}$ restricted on the plane graph $G^{-}_{C_{i}}$, respectively. Then, for any $g\in{\cal F}_{\rm inner}(G^{-}_{C_{i}})$, by the definitions of $z_{D^{\prime}_{i}-D_{i}}(g)$ (see Definition 2.1) and $z_{D^{\prime}-D}(f,g)$, we have

[TABLE]

and therefore $z_{D^{\prime}_{i}-D_{i}}(g)\geq 0$. Thus, by Theorem 3.2, all the cycles of ${\cal C}$ contained in $G^{-}_{C_{i}}$ can be reversed by

[TABLE]

flips. Hence, all cycles of ${\cal C}$ can be reversed by

[TABLE]

flips, where the last equality holds because $z_{D^{\prime}-D}(f,g)=z_{\min}(f)$ for any

[TABLE]

This completes our proof. ∎