Independent covers in outerplanar graphs - Springer Link

I n d e p e n d e n t Covers in O u t e r p l a n a r Graphs Maciej M. Systo * Institute of Computer Science University of Wroctaw Przesmyckiego 20 PL-51t51 Wroctaw Poland

Pawet Winter Institute of Datalogy University of Copenhagen Universitetsparken 1 DK-2100 Copenhagen O Denmark

ABSTRACT A subset U of vertices of a plane graph is said to be a perfect face-independent vertex cover (FIVC) if and only if each face has exactly one vertex in U. Necessary and sufficient conditions for a maximal plane graph to have a perfect FIVC are derived. A notion of an in-tree is used to study plane embeddings of maximal outerpIanar graphs (mops) and their perfect FIVCs. Finally, a linear time algorithm which finds a minimum cardinality perfect FIVC of a mop is developed. It is argued that the results are extendable to arbitrary outerplanar graphs.

1 Introduction A graph G = (V, E) is planar if it can be embedded in the plane. A plane graph is a graph already embedded in the plane. The regions defined by a plane graph are called its faces; the unbounded region is called the exterior face. The remaining faces are called interior. A boundary of a face is a cycle subgraph enclosing the face. We assume that two plane embeddings of a planar graph are equivalent if their boundary sets are the same. A planar graph G is outerplanar if it can be embedded in the plane with all vertices on the exterior face. Such embedding is called an outerplane graph, and it will be denoted by Go. G is a maximal outerTIanar graph (mop) if no edge can be added without destroying the outerplanarity. Let G be a 2-connected planar graph. With a plane embedding Gp of G there is associated a geometric dual graph Gpd, in which the vertex set corresponds to the faces of Gp, and the edge set is equivalent to the edge-adjacency relation of the corresponding faces. Removing the node v corresponding to the exterior face of G~ (together with the edges incident with v) yields the weak dual graph G~'. In order to emphasize the distinction between a plane graph and its geometric or weak dual, vertices and edges of the latter are referred to as nodes and branches respectively. *This research was partially supported by the grant RP.I.09. from the Institute of IxLformatlcs, University of Warsaw. Hospitality of the Institute of Datalogy, University of Copenhagen where this research has been completed is gratefully acknowledged.

244

A subset U C V of vertices in a 2-connected plane graph Gp is called a perfect faceindependent vertex cover (in the following abbreviated to perfect FIVC) of faces of Gp if every face has exactly one vertex in U. Note that if Gp is maximal plane, then the faceindependence of vertices is equivalent to the independence in the usual sense. A perfect FIVC in Gp corresponds in the geometric dual G~ to a set of faces .7"u which we call a perfect vertex-independent face cover (abbreviated to perfect VIFC); every vertex of Gva is contained in exactly one face of .7"rr. A perfect VIFC in G~ is nothing else than a 2-factor of G~ which consists of facial cycles. Perfect FIVCs and VIFCs have been introduced in [5] in a slightly weaker form. A subset U of vertices in a plane graph Gp is a weak FIVC if U covers all faces of Gp and at most one face of Gv contains more than one vertex. The study in [5] was motivated by the traveling salesman problem on Halin graphs. Every Hamiltonian cycle in a Halin graph H generates a set of faces 29 which covers all interior vertices in H and 7) corresponds to a weak FIVC in the weak dual graph of H which is an outerplanax graph. Yet another generalization of a perfect VIFC, known as a face cover, can be obtained by allowing faces to overlap on vertices. This notion has been recently considered by several authors, see [1] and [2]. Every plane graph has a face cover. However, as it was independently proved in [i] and [2], the face cover problem is NP-complete in general [3] Several other versions of this problem are also NP-complete. Verifying whether a plane graph has a perfect VIFC is also NP-comptete. The main subject of this paper is perfect FIVCs for mops. However, our ultimate aim is to extend our result to larger/other classes of planar graphs. In Section 2 we therefore mention necessary and sufficient conditions for a maximal plane graph to have a perfect FIVC. Subsequent sections are concerned with mops. The notion of an in-tree representing a plane embedding of a mop is introduced in Section 3. In particular, a linear time algorithm generating the boundaries of faces of an embedding specified by an in-tree is given. In Section 4 in-trees specifying identical plane embeddings are characterized, and the total number of different plane embeddings of mops is derived. Section 5 is concerned with two infinite subclasses of mops: linear and .full mops. It is shown that linear mops always have a plane embedding admitting a perfect FIVC. On the other hand, full mops have no plane embedding admitting a perfect FIVC. A linear time algorithm which finds an outerplane embedding admitting a minimum cardinality perfect FIVC is given in Section 6. Extensions of our results to arbitrary outerplanar graphs as well as suggestions for further research are given in Section 7.

2

Perfect F I V C s in M a x i m a l P l a n e G r a p h s

Let Gp = (V, E) be a maximal plane graph (i.e., no edge can be added to Gp without destroying the planarity). If Gp is Euleriaa (i.e., every vertex is of even degree), then it has a 3-coloring {V~, V2, V3} [9]. Each of subsets V~, V2, V3 forms a perfect FIVC. Consequently, in the following we turn our attention to maximal plane graphs with at least two odd degree vertices. Every vertex together with its neighbours generates a wheel in Gp. For an even degree vertex v, there are two ways to cover the faces containing v; either by taking v or by taking every second neighbour of v on the circle. If v is of odd degree, then the latter set is not independent and v must be in every perfect FIVC of Gp. Therefore no two odd degree vertices can be adjacent. However, this is not a sufficient condition for a perfect FtVC to exist.

245

a diamond

a t-path

(a)

(b) Figure 1: A d i a m o n d and a t - p a t h

A pair of face boundaries in Gp (each being a triangle) sharing an edge is called a diamond (Figure la). The vertices in a diamond which belong to only one boundary are called tervainat. It is easy to see that if Gp has a perfect FIVC U, then either both terminal vertices of each diamond D or none of them belongs to U. As a generalization of diamonds, we now introduce t-paths. They will be used to characterize maximal plane graphs with perfect FIVCs. A t-path in Gp is a sequence P = (C z, C 2, ..., C k) of distinct triangles such that: • C { and C ~+1 form a diamond in Gp for i = 1, 3, 5, ..., 2I - 1, l = [k/2j (i.e., C { shares an edge with C~+~). Let D i denote the diamond consisting of C 2~-1 and C 2i for i = 1, 2, ..., I. • The terminal vertex of D 1 (resp. D z) which belongs to C 1 (resp. C k) doesnot belong to any other triangle of P. The other terminal vertex of D 1 (resp. D z) is a terminal vertex of D 2 (resp. Dz-1). For every i = 2, ..., l - 1, a diamond D i shares its terminal vertices with those of D i-1 and D I+1. • D i and D I+1 may share at most one edge, i = 1, 2, ..., t - 1. A diamond is a t-path. Another t-path is shown in Figure lb. Let P = (C 1, C 2, ..., C k) be a t-path and (D 1, D 2, ..., D 1) be a sequence of the corresponding diamonds. The length of P is equal to k. The terminal vertex of D 1 which belongs to C 1 is called an initial vertex of P. If k is even, then the terminal vertex of D l which belongs to C A is called an end-vertex of P. If k is odd then each of the two non-terminal vertices of C A in D t is an end-vertex of P. If u is the initial vertex of P and v is an end-vertex of P, then we shall say that P connects u and v. If the initial vertex v of a t-path P belongs to a perfect FIVC U of Gp, then every terminal vertex of P has to belong to U. Furthermore, no non-terminal vertex of P can be in U. It follows that no pair of odd degree vertices in Gp is connected by an odd-length t-path. This is in fact a sufficient condition for a perfect FIVC of a maximal plane graph to exist. Suppose that no pair of odd degree vertices is connected by an odd-length t-path in Gp. To construct a perfect FIVC U in Gp we first add to U all odd degree vertices. For every even degree vertex v, we identify the closest odd degree vertex w. Ties are broken arbitrarily. As described in [6], a t-path connecting w with v can always be constructed in a maximal plane graph. If v is a terminal vertex, then it is added to U. Otherwise it is kept out of U. It can be shown [6] that under the assumption that there are no odd-length t-paths between odd degree vertices, exactly one vertex in each triangle of Gp will be in U.

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Theorem 1. A maximal plane graph Gp has a perfect FIVC if and only if no two odd degree vertices in G~ are connected by an odd-length t-path. [] 3

Representation of Maximal Outerplanar Embeddings

We now turn our attention to mops and their plane embeddings. We will subsequently use these results in connection with perfect FIVCs for mops. Note that contrary to the discussion in the previous section, we witl investigate the existence of perfect FIVCs for mops rather than for fixed outerplane embeddings. The weak dual of an outerplane graph Go is a tree; it will be denoted by T~ (or simply T if there is no danger of confusion). Each branch intersects exactly one interior edge in Go. The nodes of degree 1 in T are called external. The remaining nodes are called internal. In general, we follow the terminology proposed in [4]. Let G = (V, E) be a mop, 1VI > 3, and let Go denote its outerplane embedding. Let W denote the nodes in the weak dual T of Go. For any node v E W, let F~ denote the face of Go corresponding to the node v, and let Co~ denote its boundary" cycle. Any partially directed tree obtained from T by: • selecting an arbitrary t E W as a root, • directing some of the interior branches of T toward t, is called an in-tree rooted at t. tt will be denoted by 7~. It is shown in [7] that every plane embedding Gp of a mop G can be specified by an appropriate (but not unique) in-tree 2g. Conversely, any in-tree 2F that can be obtained from T (by choosing a root and by directing some of the branches) represents a plane embedding @ of G. Since the manner in which a plane embedding is obtained from a given in-tree is essential for subsequent results, it is given here in full. The construction was originally described in [7]. The plane embedding G~ associated with 2~ is constructed gradually. Furthermore, as Gp is constructed, boundaries of interior faces of G~ are associated with appropriate nodes of T. The boundary of the exterior face is associated with a dummy node r. Draw first the boundary C~. Place the node t within Cot. Place the dummy node r outside of Cot. Let C~ and C~ denote (temporary) boundaries of respectively the interior and the exterior face of Gp. Traverse/~ from t toward the external nodes. Let (a, b) denote the edge of Go intersected by just traversed branch (v, u), u being closer to the root (Figure 2). If (v, u) is directed (toward u), draw Co~ such that it is within C~, and does not enclose the node u. This is always possible since C~' and C~ share just one edge (a, b). Place v within the new boundary, denoted by C~. Update C~ by replacing (a, b) by the sequence of edges connecting a and b in CoL If (v, u) is undirected, let g denote the first node on the path from v to the root which has a directed branch entering it. If no such branch exists, let 0' = r. Draw Co~ such that it is within C[, and does not enclose the node g. This is always possible since Co~ and C~ share just one edge (a, b). Place v within the new boundary denoted by C~. Update C[ by replacing (a, b) by the sequence of edges connecting a and b in C2. In either case, the planarity of the drawn graph is preserved; (a, b) is the only edge of Co~ common with the graph drawn so far. Furthermore, upon completion, nodes of the

247

c~~b• e@r

d

b

c

g Figure 2: In-tree and the corresponding embedding

in-tree T placed in Gp correspond to different internal faces of Gp. The dummy node r corresponds to the external face. The above construction suggests a very simple linear time algorithm identifying all boundaries of the embedding Gp corresponding to a given in-tree T rooted at a node t.

procedur e FACES (T, t, Co, C~) ; begin C~:=C~:=CI; t:=r; S:={zeTiz adjacent to t); while S # @ do begin v:=any element in S; S:=S\{v}; u:--node adjacent to v and on the path to t; ~:= if (v,u) is directed then u else 5; v._

v,

~.--

~

V

Cp.-C o , Cp. -Cp~C o ;

S:=S U {z6Tlz adjacent to v, z#u}; end end It is assumed that the boundaries of Go are given as lists of their edges, and they are in turn collected into a list denoted by Co. Similarly, the boundaries of G~ will upon completion be represented by lists of their edges. They will be collected into a list denoted by Cp. The dummy node corresponding to the exterior face of Gp is denoted by r.

4

N u m b e r of Embeddings

Our next result will show that there are 2/-2 different embeddings of a mop G with f interior faces. If G has n vertices and m edges, then f = m - n + 1. If we fix a root t of T, then there are exactly 2 ]-1 different in-trees rooted at t; T has f - 1 branches, each branch can be directed (toward the root) or it can be left undirected. We define a transformation ¢ of an in-tree T rooted at t which preserves the set of boundaries by replacing each directed (resp. undirected) branch incident with t by an undirected (resp. directed toward t) one. It is easy to check that in-trees T and ¢ ( T ) generate the same set of faces.

248

L e m m a 2. If T and T ' are two different in-trees rooted at the same node t, then T and T ' generate different sets of faces unless O(T) = T'. P r o o f . Let us consider a branch b closest to the root t which has a different orientation in T and in T'. If b is incident with t and T ' # ¢ ( T ) , then either there exists another branch b' incident with t which has the same orientation in T and in 2n', or there is no other branch incident with t. In the former case, the in-tree in which b' and b have the same orientation corresponds to an embedded graph with a face whose boundary contains both edges intersected by branches b' and b. In the other in-tree, no such face exists. In the latter case, we choose a second-closest branch with different orientation in 2n and in T' and proceed to the general case when b is not incident with t. Let b' denote the first branch on the path from b to t. The branch E has the same orientation in and in 2n'. Let e and d denote the edges in G intersected by b and br, respectively. Both when E is directed and undirected, one can easily check that in only one of the embeddings specified by T and T', there exists a face whose boundary contains both e and e'. [] It follows that all in-trees of T rooted at any node t produce 2 f-2 different plane embeddings of G. Our next goal is to show that there are no other plane embeddings of G. L e m m a 3. Let T~ be an in-tree rooted at s, s # t. Then there exists an in-tree T rooted at t generating the same set of faces. P r o o f . Let 2n' be an in-tree of T rooted at s. If 8 = t, then we are done. Otherwise, T ' can be transformed to an in-tree 2~ rooted at t by reorienting some of the branches of 2~' on the path ~r from s to t. For this purpose, we apply along the path 7r the transformation ~ which does not change the set of faces. More precisely, let us assume that ~r = (Vo = s, 731,732,..., ?3k ~--- t). ~/~ iS obtained from T ' by applying the transformation ~*: p r o c e d u r e ~* ( ~ , ~') ; begin s:=first node in ~; t:=last node in ~; while s ~ t do begin p:=second node in ~; if (p,s) is directed toward s then ~ : = ~ ( ~ ) ~':=~-\{s}; ~ :-'-p; end end

It is easy to check that the resultin~ tree T is an in-tree rooted at t, and by Lemma 2, generates the same set of faces as T'. [] T h e o r e m 4. A mop G has exactly 2y-2 different embeddings which can be generated by the maximal family of in-trees rooted at any node t of T which is complement-free (i.e., if 7n belongs to the family, then ~(7~) does not belong).

249

(a)

(b)

(c)

(d) Figure 3: Mops without perfect FIVCs

5

Full and Linear Mops

In this section we introduce two infinite subclasses of mops and discuss their properties with respect to perfect FIVC. A mop is said to be full if all internal nodes of its weak dual T have degree 3. A mop is said to be linear if all internal nodes of its weak dual have degree 2, i.e., when the branches form a path. L e m m a 5. If a mop is full, then none of its plane embeddings has a perfect FIVC. P r o o f . The proof is by induction on the number no of internal nodes in the weak dual T of Go. For no = 1, it can be easily verified by inspection that the unique full mop has no embedding for which a perfect FIVC exists (Figure 3a). Suppose that the result is true for all full mops with n _< no internal nodes. Consider a fuI1 mop G with n = no + 1 internal nodes. Assume that G has a plane embedding Gp for which a perfect FIVC, denoted by U, exists. Let T denote the in-tree specifying this embedding. It is always possible to find an internal node v E T, other than the root, and being adjacent to two non-root external nodes x and y. Let a, b, and c denote the three vertices on the boundary C~ in Go. Furthermore, let d and e denote the vertices, other than a, b, or c, on the boundaries Co~ and Cov, respectively (Figure 3b). There are three cases to consider. • c E U. The mop G" obtained by deleting d and e is full. U is also a perfect. FIVC of Gp. This contradicts our inductive hypothesis. H

• a E U. Then there is no way to cover the face F~, a contradiction. By a similar argument, b E U will lead to a contradiction.

250

• a, b, c • U. In order to cover faces F~ and Fpv, {d, e} _C U. Thus, o n e of the branches (x, v) or (y, v) must be directed toward v while the other must be undirected. Assume that (x, v) is directed toward v. Remove vertices d and e from Gp, and from U. The only uncovered faces of the reduced embedding G~ are the two faces containing vertices a, b, and c. By adding c to U, a perfect FIVC for the reduced full mop G" is obtained. This contradicts our inductive hypothesis. [] The proof of Lemma 5 shows in fact that if an arbitrary mop G has no perfect FIVC, then neither a mop obtained from G by adding three vertices as shown in Figure 3d has a perfect FIVC. Unfortunately, we cannot characterize mops without perfect FIVCs in terms of minimal forbidden subgraphs. For instance, mops shown in Figures 3a and Figure 3b do not have perfect FIVCs since they are full. If we delete any vertex of degree 2 in the mop shown in Figure 3b, we obtain a mop which has a perfect FIVC, and which has a mop shown in Figure 3a as its subgraph. Finally, we remark that there are non-full mops without perfect FIVCs. It can be verified by inspection that the mop shown in Figure 3c has no perfect FIVC. L e m m a 6. Every linear mop has a perfect FIVC. P r o o f • The lemma holds if G has only three vertices. We assume in the following that G has at least four vertices. The weak dual T of Go is a path• Number its nodes 1 , 2 , . . . n , beginning at one of the end-nodes. Let A, B, C, D denote subsets of vertices in Go. Consider the following procedure PARTITION p r o c e d u r e PARTITION(G,T) ;

begin

A:={v,}; B:--{v2};c:--{v3};o:--{v4}; f o r i:=3 step

I until

begin Vq : =C i \ C 4 - ' -o%-o ; • __Ci-2 \ Ci-1 . Vp •--vo \vo add vq to t h e

subset

n do

containing

vp;

end end

Initially, when PARTITION is called, A, B, C, D contain one vertex from Co1 [3 Co2 = {Vl, v2, v3, v4} each. We will show that upon completion these subsets specify a partition of all n + 2 vertices in Go. At least two of these subsets will yield a perfect FIVC for the embedding Gp corresponding to the in-tree f with node 1 as its root, and with all branches directed toward the root. Each vertex of Go will be in at most one subset. This follows directly from the fact that during the i-th iteration, i >_ 3, a uniquely determined vertex in C~o\C~o-1 is placed in some subset. In fact, each vertex of Go belongs to one of the subsets A, B, C, D. Suppose that this is not the case. Let Co~ denote a boundary containing a vertex vq not in any of these subsets. Assume that i is smallest possible. Since i > 3, vv 6 C~-2 \ C~-1 belongs to one of the subsets, and vq should have been placed in the same subset as vp, a contradiction. We have a partitioning of vertices of a linear mop. It can be easily verified that a pair of vertices in the same subset is separated by at least one face. Furthermore, four vertices on two consecutive boundaries are in mutually different subsets (Figure 4).

251

C

A

C

Figure 4: Partitioning of a linear mop C 1 and Con have vertices in three different subsets. Consequently, there are at least two subsets containing vertices on both Co1 and Co% Suppose that A is one of these subsets. A is perfect FIVC for the plane embedding Gp specified by the following in-tree T: node 1 is the root, and all branches are directed toward the root. All boundaries in Gp, apart from the exterior boundary C~ (where r is the dummy node) and C~, consist of four vertices belonging to a diamond in Go. Vertices on these boundaries axe therefore in different subsets. Thus, these boundaries are certainly covered by exactly one vertex in A. Furthermore, Cp and C~ are covered by exactly one vertex from A merely due to the manner in which A was selected. [] 6

Perfect

FIVCs

for mops

In this section we describe a linear time algorithm which finds a plane embedding Gp of an arbitrary mop G admitting a perfect FIVC, or decides that no such embedding exists. If there are several embeddings admitting perfect FIVCs, then the algorithm finds one admitting the minimum cardinality perfect FIVC. In addition, the algorithm determines this perfect FIVC. Let T denote a weak dual of an outerplane embedding Go of a 2-connected mop G. Select one of the external nodes of T as a root denoted by t. To each node v E T, v # t, attach as many dummy branches (and therefore also as many dummy end-nodes) as there are edges on Co~ which also belong to the exterior face of Go. Assume that the dummy branches intersect their respective edges. Consider an arbitrary node v other than the root t (either in T or a dummy node). Let (v, u) denote the branch incident with v, and such that u is on the path from v to t. Let (a, b) denote the edge of Go intersected by the branch (v, u). Assume that a is to the left and b is to the right of (v, u) when looking from v toward u. Let G ~ consist of the edge (a, b) if v is a dummy node. If v is any other node, let G ~ denote a subgraph of Go consisting of the union of the edges on the boundaries Co~, x = v or x is "behind" v when looking from u toward v. Define the following minimum cardinality covers found among all plane embeddings of G ~ rooted at v: IV= FIVC of all but the exterior face of G ~, LV= perfect FIVC of G v with the exterior face covered by a,

252

R ~ = perfect FIVC of G ~ with the exterior face covered by b, EV~- perfect FIVC of G" with the exterior face covered by a vertex other than a and b. These covers are defined for each d u m m y node v as follows. I v = 0, L v = {a}, R" - {b}, while E" is undefined. A union of an undefined set with any other set yields an undefined set. Any node for which the above four covers are determined is said to be labeled. Hence, all d u m m y nodes are initially the only labeled nodes. W'e show next how to obtain covers for unlabeled nodes. If there exists an unlabeled node v, v ~ t, it can always be chosen such that two of its three neighbours are labeled. Let u denote the unlabeled node, and let x and y denote labeled nodes. Assume that in order to reach x (resp. y) from u, one has to turn left (resp. right) at v. Given I x, L x, R ~, E x and I y, L y, R y , EY, the recurrence rules for I v, L '~, R v , E v can be determined by a straightforward case analysis (of all possible ways of orienting the branches (x, v) ~nd (y, v)). A detailed proof of the validity of these rules can be found in [8]. L e m m a 7. L ~ = I x U L y, R v = R ~ U I y . Furthermore, I v and E v are the minimum cardinality sets chosen from the following families:

I v e { E x u I ~ , I ~ u E v} E" E {L~U R ~ , E ~ U E y} In order to make computations in linear time, and in order to be able to keep track of the solutions found so far, the following information is maintained for each cover of G v. • pointers to the appropriate covers in G x and G y whose union yielded the cover in

G~ • cardinality of the cover (equal the sum of cardinalities of its two subcovers; the only exception occurs if E v = L ~ U R y where L ~ and R ~ share a vertex), • orientation of branches (x, v) and (y,v), in order to identify the underlying plane embedding. Note that the covers are represented by binary trees with cover-elements (i.e., vertices of G) as their leaves. Forming a union of two covers is equivalent to attaching two binary trees to a common new root. After n - 1 iterations all nodes but the root t are labeled. Let x denote the node adjacent to t. Let c be a vertex in Co~ but not in Cy. Again, by a straightforward case analysis [8], the recurrence rules for the minimum cardinality perfect FIVC can be easily derived. L e m m a 8. The minimum cardinality perfect FIVC U of G is the smallest set among

{I u {c}, L

D

Once the smallest of the above three covers has been determir.ed, its vertices can be identified by traversing the binary tree structures for the subcovers As these binary trees are traversed, the orientation of branches of the in-tree specifying the optimal embedding is retrieved. T h e o r e m 9. Minimum cardinality perfect FIVC (and the underlying plane embedding) of an arbitrary mop can be found in O(n) time. []

253

7

Extensions

and

Suggestions

for Further

Research

The notion of an in-tree for a mop can be easily extended to arbitrary 2-connected outerplanar graphs. There seems to be at least two different ways to do it. One is to consider the duM of the outerplane embedding. Its nodes can be incident with any number of branches. On the other hand, degrees of nodes in the weak dual for a mop are exactly three. Arguments similar to those for mops can be used to prove that to each plane embedding of an outerplanar graph correspond two in-trees, T and ¢(T), rooted at some a priori chosen non-pendant node. Another approach is to add dummy edges to the outerplanar graph so that it becomes maximal. A linear time algorithm for this kind of fill-in can be found in [10]. Faces in the outerplane embedding with more than three edges on their boundaries will be split up into small faces with three-edge boundaries. Consider a weak dual of such a mop. Plane embeddings of the original outerplanar graph can be represented by in-trees for the expanded mop. The only restriction is to keep the branches intersecting dummy edges undirected. Given an in-tree of an expanded mop, boundaries of its faces can be identified in linear time by the procedure FACES defined in Figure 3. The boundaries of the faces of the corresponding embedding of the original outerplanar graph- can be obtained by successive pairwise "splicing" of boundaries containing dummy edges. This can be done in constant time if appropriate data structures are used. The linear time algorithm for minimum cardinality perfect FIVC of a mop can be extended to arbitrary outerplanar graphs [8]. If appropriate data structure are used, it is possible to preserve the linearity of the algorithm. A problem which so far remains unsolved is a complete characterization of mops which have a plane embedding admitting a perfect FIVC. Also the problem of identifying "good" (but not necessarily perfect) FIVCs for mops as well as arbitrary outerptanar graphs is of interest. The dynamic algorithm presented in this paper could most likely be extended to cover this more general case. Another avenue of research related to the representation of, plane embeddings would be an extension of the notion of in-trees to 2-trees and 2-connected partial 2-trees (seriesparallel graphs). Although 2-trees have a more complicated structure, the problem of characterizing all plane embeddings by means of appropriately defined "pseudo-dual" graphs seems to be tractable. A paper concerning this subject is under preparation. Other "regular" classes of planar graphs could be investigated as well (e.g., Halin graphs or planar subclasses of a-trees). The impact of our work on the representation of plane embeddings of arbitrary planaz graphs remains to be investigated.

References [1] D. Bienstock and C.L. Monma, On the complexity of covering vertices by faces in a planar graph, S I A M J. on C o m p u t . 17 (1988) 53-76. [2] M. Fellews, F. Hickling and M.M. Systo, Topological parametrizagion and hard graph problems (extended abstract) Washington State University (1985). [31 M. Fellows, Personal communication (1986). [4] A. Proskurowski and M.M. Systo, Minimumn dominating cycles in outerplanar graphs, I n t e r n a t . J. C o m p u t . Inform. Sci. 10 (1981) 127-139.

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[5] M.M. Systo, On two problems related to the traveling salesman problem in Halin graphs, in: G. Hammer and D. Pallaschke (eds.), Selected Topics in Operations Research and Mathematical Economics, Lecture Notes in Economics and Math. Systems 266, Sprlnger-Verlag, Berlin (1984) 325-335. [6] M.M. Systo, Independent face and vertex covers in plane graphs, TR N-184, Institute of Computer Science, University of Wroclaw (1987). [7] M.M. Systo and P. Winter, Plane embeddings of outerplanar graphs, Technical Report, Institute of Datalogy, University of Copenhagen (1988). [8] M.M. Systo and P. Winter, Fane-independent vertex covers of outerplanar graphs, Technical Report, Institute of Datalogy, University of Copenhagen (1988). [9] R.J. Wilson, Introduction to Graph Theory, Longman, London (1972). [10] P. Winter, Generalized Steiner problem in outerplanar graphs, B I T 25 (1985) 485496.