Lemma 1. A 3-connected maximal generalized outerplanar graph is a wheel.

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1 122 (1997) MATHEMATICA BOHEMICA No. 3, 225{230 A LINEAR ALGORITHM TO RECOGNIZE MAXIMAL GENERALIZED OUTERPLANAR GRAPHS Jo C cere, Almer a, Alberto M rquez, Sevilla (Received November 16, 1994, revied May 16, 1996) Abtract. In thi work, we get a combinatorial characterization for maximal generalized outerplanar graph (mgo graph). Thi reult yield a recurive algorithm teting whether a graph i a mgo graph or not. Keyword: outerplanar graph, generalized outerplanar graph MSC 1991 : 05C10, 05C75 1. Introduction The main concept of thi paper wa introduced by Sedl ek in [6]. He denedgeneralized outerplanar graph a graph with a planar repreentation uch that, at leat, one end-vertex of each edge lie on the outer face. Alo, he gave a characterization in term of forbidden ubgraph (ee Figure 1). Clearly, thi i a way to generalize the well-known concept of outerplanar graph. Thee two kind of graph have been ued in the deign of printed board where it i required that all terminal (or one end-terminal of each wire) be placed on the periphery of the chip of the board [3]. Of coure, it would be very ueful to get linear algorithm for recognizing outerplanar and generalized outerplanar graph. The former wa obtained by Mitchell in [4], and, in thi paper, we preent a linear algorithm for the recognition of maximal generalized outerplanar graph ince a tet whether a graph i generalized outerplanar or not, eaily follow from our algorithm. A maximal generalized outerplanar (mgo) graph i a generalized outerplanar graph uch that no edge can be added without violating thi property. 225

2 G 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8 G 9 G 10 G 11 G 12 Figure 1. The forbidden ubgraph of Sedl ek We will ue [1] for the common graph notation, except for uing the term vertex intead of point, and edge intead of line. Nonethele, let u recall ome helpful concept. A graphgi2-connected when at leat two vertice ofgmut be removed to diconnected it; if there are two vertice,uandv, ofguch that their removal diconnect G, we call them a eparation pair. The graph induced by u, v and the vertice of the connected component ofg {u,v} i called aplitgraph. A 2-connected graph with no eparation pair i aid to be 3-connected and a maximal 3-connected ubgraph i called a 3-component. Alo, we mention a pecial kind of graph. Forn >4, thewheelw n i dened to be the graphk 1 +C n?1. In [7], Tutte howed that every 3-connected graph either i a wheel or can be built by a equence of the following two operation overw n : 1. Add a new edge. 2. Replace a vertexw having a degree at leat 4 by two adjacent verticew 0 and w 00 uch that each vertex formerly joined towi joined to exactly one ofw 0 and w 00 o that in the reulting graph,w 0 andw 00 have a degree at leat The reult 226 The previou reult yield Lemma 1. A 3-connected maximal generalized outerplanar graph i a wheel.

3 Proof. If we try to contruct a mgo 3-connected graph from a wheel by applying the two Tutte' operation then we realize that we are jut allowed to add an edge between two non-conecutive vertice of the periphery of the wheel but thi new graph i not generalized outerplanar becaue it contain a ubgraph homeomorphic to the forbidden ubgraphg 10 (ee Figure 1). On the other hand, the only vertex with a degree at leat 4 i the center of the wheel, o we can apply operation 2 jut to thi vertex. But, again, we obtain a nonplanar graph or a graph which contain a ubgraph homeomorphic to the Sedl ek ubgraphg 11 (ee Figure 1). After we have characterized 3-connected mgo graph, it i traightforward to check the characterization in the 2-connected cae. Lemma 2. The only 2-connected maximal generalized outerplanar graph without 3-componentiK 3. Proof. There exit only one 2-connected graph with 3 vertice:k 3, it ha no 3-component and it i mgo. So, conider a graph G with at leat four vertice. In an outerplane embedding ofg, if all vertice lie in the exterior face then the graph i outerplanar, and it i eay to check that an outerplanar graph with at leat four vertice cannot be mgo. Thu, there exit a vertex which doe not lie in the exterior face but thi vertex mut be joined only with the verticev 1,...v n of the exterior cycle (every edge ha an end-vertex on the exterior face). Without lo of generality, we can uppoe that the verticev 1,...v n are conecutive in the cycle. Now, if the edgev 1 v n exit, then the graph ha a 3-component and if the edgev 1 v n doe not exit, thengi not maximal becaueg+v 1 v n i generalized outerplanar. So, there are no graph with at leat four vertice under the condition of the lemma, and we have the proof. Now we are ready to olve the general cae. Theorem3.Let{u,v}beaeparationpairofa2-connectedgraphGthatplit thegraphing 1,...,G p.giamgographifandonlyifthefollowingfourcondition are atified: 1.uvianedgeofG 1,...,G p. 2.G 1,...,G p aremgograph. 3.AtmottwoofthecomponentG 1,...,G p arenotiomorphictok 3. 4.EachG 1,...,G p haageneralizedouterplaneembeddinguchthattheedgeuv belong to the exterior face. 227

4 Proof. Aume thatgi a 2-connected graph and{u,v} i a eparation pair. Let Z be the exterior cycle of a generalized outerplane embedding of G. Since thi cycle connect the graph, the verticeuandv belong toz. Ifuandv are conecutive inz then the edgeuv exit, otherwie, ince the endvertice of every edge are in the ame plit graph, the maximality ofgimplie that the edgeuv belong togand o, it belong tog 1,...,G p (condition 1). Condition 2 follow from the fact thatg 1,...,G p inherit the mgo property ofg. By virtue of Lemma 2, all but at mot two of the componentg 1,...,G p are iomorphic tok 3. On the other hand, ifg 1,G 2 andg 3 are dierent fromk 3 then G 3 mut be embedded in an internal face of eitherg 1 org 2 and o,gwould not be generalized outerplanar (condition 3). Clearly,uandv lie on the exterior face of the embedding ofg i (1 6i6p) induced by the generalized outerplane embedding ofg. Ifuandvare not conecutive in the exterior cycle then we can plitg i into a new 2-connected component and ak 3. Thu, uv belong to the exterior cycle and condition 4 hold. Converely, we build the generalized outerplane embedding of G in the following way. Conider two component,g 0 andg 00, uch that they are not imultaneoulyk 3 (ee condition 3), and it generalized outerplane embedding uch that uv belong to the exterior cycle (ee condition 4). Merging their planar repreentation, we obtain a generalized outerplane embedding ofg 0 G 00 and alo, ifg 0 andg 00 are maximal then thi new graph i maximal a well. Now, we can join component iomorphic tok 3 loing neither the maximality property nor the generalized outerplanarity property. 3. The linear algorithm Theorem 3 i the reult we need to deign a recurive algorithm for teting whether a graph i mgo or not. Roughly peaking, the algorithm work in the following way: it plit the input graph into two 2-connected component, check condition 1 and 3 of Theorem 3 and recurively ue thee component a input. During the backtrack of the algorithm, condition 4 i teted and the recurion end when the ituation of Lemma 1 or Lemma 2 occur. One important tep of the algorithm i to nd the 3-component of the input graph. The linear algorithm of Hopcroft and Tarjan (ee [2]) can be ued to do thi. Alo, in [5], the author chooe to explore the graph by uingdepth-firtearch (DFS) and thi i the method that our algorithm ue. 228

5 MGO-TEST Algorithm. Let G be a 2-connected graph with M vertice having a lit of verticev and edgee. Let u uppoe that all vertice are labelled with `interior'. Step1: If E >3M 6 thengi not mgo and top. IfG=K 3 thengi mgo and top. Otherwie: Step 2: Uing DFS, check whether G i 3-connected. 1.G i 3-connected. Check whethergi a wheel (the degree ofm 1 vertice i 3). If it i not thengi not mgo and top. Ele: (a)g=k 4. G i mgo if and only if there exit a vertex labelled `interior'. Stop. (b) G K 4.G i mgo if and only if the vertex with a greater degree i labelled `interior'. Stop. 2.G i not 3-connected. Look for a eparation pair{u,v} ofg. Ifuv/ E then G i not mgo and top. Ele, build plit graphg 1,...,G p labellinguandv a `exterior' in eachg i. Except one or two, all the graphg 1,...,G p mut bek 3, orgi not mgo and top. IfG 0 andg 00 are uch graph thengi mgo if and only if bothg 0 andg 00 are mgo. Step 1 enure that we are not in a trivial cae. In 2.2, we dene a recurion loop that plit the input graph, check whether there are the correct number ofk 3 and delete them. The condition for ending thi recurion i in 2.1 where the algorithm check whether we have a wheel and whether it center can be placed not in the exterior face. Theorem4.Totetwhetheragraphimgoornot,needtimeO(n)withthe MGO-TEST algorithm, and thi i optimal. Proof. Clearly, the tep that dominate the calculation i to check by DFS whether the graph i 3-connected. The cot of thi tet io(n), which complete the proof. Reference [1] F. Harary: Graph Theory. Addion Weley, Reading Ma., [2] J. E. Hopcroft and R. E. Tarjan: Dividing a graph into triconnected component. SIAM J. Comput. 2 (1973), 135{158. [3] M. C. van Lier and R. H. J. M. Otten: C.A.D. of mak and wiring. T. H. Rept. 74-E-44, Dept. Elect. Engrg. Eindhoven Univerity of Technology. [4] S. Mitchell: Linear algorithm to recognize outerplanar and maximal outerplanar graph. Inform. Proce. Lett. 9 (1979), 229{232. [5] T. Nihizeki, N. Chiba: Planar Graph: Theory and Algorithm. North-Holland, Amterdam,

6 [6] J. Sedl ek: On a generalization of outerplanar graph. aopi P t. Mat. 113 (1988), 213{218. [7] W. T. Tutte: A theory of 3-connected graph. Indag. Math. 23 (1961), 441{455. Author' addree: Jo C cere, Geometr a y Topolog a, Univeridad de Almer a, Almer a, Spain, jcacere@ualm.e; Alberto M rquez, Matem tica Aplicada I, Univeridad de Sevilla, Sevilla, Spain, almar@obelix.cica.e. 230