On the depth of a planar graph

ON THE DEPTH OF A PLANAR GRAPH A graph G is said to be plamr when it is isomorphic to a graph G(n) whose vortex cet V is a paint set in a plant R while the edges ;u-e Jardan curves in R such that two different edges have, at E?GS~.end points in ccmnmn. A diagram of 3 planar graph C, which ~mfcmm with these conditions, is calted ;1plmar represent;rtion of G. TWOplanm=rcprestmtzrtisns of G will br qqrded as distinct if they cannot be made to coinwide with one another by tlastic deformation of the plane. Irr this paper, only finite pkmar graphs witt be considered. A graph G is Said to be simpk if it h;zs neither loops nm multipIe edges. We write G = iI Fe E), where V is the set of vertices of G and E is the set of edges af G. A graph 6’ with p vertices and q edges is called a (_P,@-graph. For the definitians c>fa rtlgion of a planar graph in ;I planar representation, the boundary of a region, an interior (finite) regian, the exterior (infinite) region, etc., see f 2 1. Let G be a graph. Let s4 be the sot of all distinct planar representatims of G. d is a finite set. Let T E ~4 and let FT be the cxtorisr region of G in T. Let u E I/ and (3~= {R ; R is a region of 7’ ;md u lies on the baundrtry of R). Let d(R, S) denote the length of the shortest path between the vertice R and S of T*, where T* is the pImar geometric dual of T(R and S are r&ons of 7”). 63 k hit ion 2 . T he depth zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA of G in ?‘l denoted by dcp#), is defined as follows: dep,(C) = max dq+(u). zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG VEY 1 Definiition 3. We define the depth of C as fbknvs and denote it by dep(Gj: dep(G) = min dep,(G). zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED TEA A graph G is said to be oute~lanar if it can be embedded on ff plane that all the vertices of G lie on the boundary of th,e exterior region. unress otherwise stated, we will consider only simple graphs. Rerrrrirk. ‘The following are the immediate consequenms of t,re above de fhitio ns and [ 1, Theorem 11.9, Corollary 11.91, (aj If G is disconne&d and G,, . . . . G, (t > 2) are the connected components of G, then dep(G) = maxi=,_ t dep(G& Cb) A graph G is outerplanar if and only if dep(G) =r:0. (cj Let G be a @&graph. If q > 2~ - 3, dcp(G) ;;SrI. (dj If the degree of each vertex of a @,&-graph G is at least 3, then dcp(Gj 2 1. cCe, If the de gree of each vertex of a (rj.q)-graph G is odd and Q = 2p --3, then dep(C) > 1. if) If c @&graph G has at most one vertex of degree 2 and q = 2p ---3, then dep(G) > 1. (g) tf G’ is a subgraph of C, dep(G”) < dep(C). Now we generalize the notion of the outerplanar graph. Definition 4. Let G be a @&-graph with p > 3. G is called k-auterpiarrar k G p) if it can be embedded on the plane so that k vertices lie OII the boundary of the exterior region and it cannot be embedded on the plane so that more than k vertices he OVA the boundary of the exterior region. itian 5. ALk-outerpfanar (p&-graph G is carted muxin?al k-outerif nto edge can be added to G without losing k-outerf:lanarity. is a k-~tr rerplanar (p,qj-graph and if k = pC,then G is 084terpk~~r. .= 3, thiet76’ is a Mrasimal pklnaP gFfq?h. K G. Kurw, s.K. blastt / an titc &pth zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK of a zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON plur1 ur graph 65 zyxwvutsrqponml hoof. Ce zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA I. Let G be a fcmst or a trm. W ere q G p -- I. G has only one region, namely the exterior region, whose boundary has q edges. Since q < p - I, it fcNows thut q < zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE 3p - 3 - q. Ckw 2. Let C trave s1circuit. So any region of G has at least 3 edges belonging to its boundary, auzh of which belongs to some circuit of G. Let, if possrble, G have a region R with IE= 3p . 3 -- q + t edges on its boundary, where t is a positive integer. Lzt 3p _- 3 - q + d = x ,+y, where s denotes tht: number of edges on the boundary of R belonging to Some circuit of G and _I*denotes the number of edges on the boundary of R not beiong~ng to any circuit af G. G has Y= q - p + 2 regions. So, q 3 qCx + 3(q -- p + 1)) + I*.Thus we see that t < 0, which is a contradictian. Hence the Emma follows. Lemma 2, If G b u maxhal k-ntrterphnar @,&-graph, then q = 3p -- k --~3. PtooO: In a maximal k-outerplanar graph, there is only one region with k edges on its boundary and the other regions have 3 edges each. Eience 4= j(k+3(q_p+i)).Soq=3p--k--3. Proof. Let 7% SB and T* be the planar geometric dual of T. Let 93 = (S: d(FTI S) = 1, S is a vertex of TP]. if R ;f F), then it is clear that d(FT, R) = 1 + min SEca d(S,R), Now dep(G,) a d. Hence dep#*) 2 d. for SQIII~” u* E I/. It is easy ta see that, for any u E V, min deprl(u) = nrin d(S, RS, /?EilvSG BJ {FT) where au = {R: R is a region of T and v lies on the boundary of R} and d(S, R) denotes the length af the shortest path between S and R in I? Hence d(S, R) 2 d, for any R E dv* and every S E 93 zyxwvutsrqponmlkjihgfedcbaZYXWVUT u {I $ }. N ow , if R E au*, d(E$, Rf = 1 + min FEQ!? d(F, R) >‘ d + 1. ihvr dep(C) 2 s,,,,. The abow two corol~laries follow easily from ‘2~w9rcni 1. Again. putting A-= 3 in Corollary 1.3, we get: Let G be a maximal planar graph with p vertices, where p = 3[2’), for some non-negative integer s. Sinc:e C is t? maximal pianar graph, by Corvllary I .4, dq..( 0 2 s. Now we proceed to C0nstruct a maximal planar graph with 3(Y) vertices whose depth is exactly equ;~l to S. zyxwvutsrqponmlkjihgfe C’mstrwtim. Draw a polygon with p = 312’) vertices. Along the polygon, label the vertkos aI, u2, . . . . ap. Triangulnte the polygon by joining Q1to Q3, 6z3t. . . . up __t to get the graph G(D). Next join a2 to u4, u4 to u6, , up ,_2 t(.) flp. u /,, to u2 in G(O) to get G’(1). Construct G(2) from G( 1) by joininga-, to u6, a6 to ulo, . . . . up. 3 IO ul. Continue this process of joining the v&-t&s iying in the exter:or re&on of G(i) alternately, to get C(i + 1). always starting from u2. F9alIy. we wit1 get 3 graph G(s) whkh is a maximal planar graph. It can be rasily verified that this graph has depth exa&ly equal to s. Now we shall see that two maximal planar graphs with the same ! -1ber of vtirtfe‘es need not have the same depth. Using ithe construction given abavtz, we GNI construct a graph on 12 vertices of clcpth 2. which is m;rximrrlly planar. Consider the ioosahedron which tls a maximal planar graph on 12 vertices. Since this graph is un\quely drawn on the surke of ;i sphere, one can errsily verify (by using Definition 3) that this graph is of depth 3, l l l Let &I) denote the maximum depth a planar graph with 12vertices can have. Find an expression for c’(p) in terms of i>. k?ferences 1969). k. 1967