Wak Unit Disk and Intrval Rprsntation o Graphs M. J. Alam, S. G. Kobourov, S. Pupyrv, and J. Toniskottr Dpartmnt o Computr Scinc, Univrsity o Arizona, Tucson, USA Abstract. W study a variant o intrsction rprsntations with unit balls: unit disks in th plan and unit intrvals on th lin. Givn a planar graph and a bipartition o th dgs o th graph into nar and ar dgs, th goal is to rprsnt th vrtics o th graph by unit-siz balls so that th balls or two adjacnt vrtics intrsct i and only i th corrsponding dg is nar. W considr th problm in th plan and prov that it is NP-hard to dcid whthr such a rprsntation xists or a givn dg-partition. On th othr hand, w show that sris-paralll graphs (which includ outrplanar graphs) admit such a rprsntation with unit disks or any nar/ar bipartition o th dgs. Th unit-intrval on th lin variant is quivalnt to thrshold graph coloring, in which contxt it is known that thr xist girth-3 planar graphs (vn outrplanar graphs) that do not admit such coloring. W xtnd this rsult to girth-4 planar graphs. On th othr hand, w show that all triangl-r outrplanar graphs and all planar graphs with maximum avrag dgr lss than 26/11 hav such a coloring, via unit-intrval intrsction rprsntation on th lin. This givs a simpl proo that all planar graphs with girth at last 13 hav a unit-intrval intrsction rprsntation on th lin. 1 Introduction Intrsction graphs o various gomtric objcts hav bn xtnsivly studid or thir many applications [17]. A graph is a d-dimnsional unit ball graph i its vrtics ar rprsntd by unit-siz balls in R d, and an dg xists btwn two vrtics i and only i th corrsponding balls intrsct. Unit ball graphs ar calld unit disk graphs whn d = 2 and unit intrval graphs whn d = 1. In this papr w study wak unit ball graphs: givn a graph G whos dgs hav bn partitiond into nar and ar sts, w wish to assign unit balls to th vrtics o G so that, or an dg (u, v) o G, th balls rprsnting u and v intrsct i th dg (u, v) is nar and do not intrsct i th dg dg (u, v) is ar. Not that i (u, v) is not an dg o G, thn th balls o u and v may or may not intrsct. W rr to such graphs as wak unit disk (d = 2) and wak unit intrval graphs (d = 1). A gomtric rprsntation o such graphs (particularly, a mapping o th vrtics to unit balls in R 2 or R), is calld a wak unit disk rprsntation or a wak unit intrval rprsntation; s Fig. 1. Nar dgs ar shown as thick lin sgmnts and ar dgs ar dashd lin sgmnts and w us this convntion to distinguish nar/ar dgs in th rst o th papr. Unit disk rprsntations allow us to rprsnt th dgs o a graph by spatial proximity, which is intuitiv rom th point o viw o human prcption. Wak unit disk graphs also allow to arbitrarily orbid dgs btwn crtain pairs o vrtics, which is usul in rprsntation o almost unit disk graphs. It has bn shown that wak unit intrval graphs can b usd to comput unit-cub contact rprsntations o planar graphs [5, 18]; s Appndix A.
a g a c b c g b (a) d d a b c d g i h j l n o k m p (b) a b c h i j l n (c) d g k o p m Fig. 1. (a) A graph with an dg-labling and its wak unit intrval rprsntation. (b-c) A graph with an dg-labling and its wak unit disk rprsntation. In th igurs w indicat nar dgs with solid lins and ar dgs with dashd lins. Unit disk graphs hav bn xtnsivly studid or thir application to wirlss snsor and radio ntworks. In such a ntwork ach snsor or radio can b modld as a dvic with a unit siz broadcast rang, which naturally inducs a unit disk graph by adding an dg whnvr two rangs intrsct. This stting maks it asy to study various practical problms. For xampl, in th rquncy assignmnt problm th goal is to assign rquncis to radio towrs so that narby towrs do not intrr with ach othr [15]. A waknss o th unit disk modl is that it dos not allow or intrrnc btwn nods (.g., du to gography) and it dos not account or th possibility that a pair o nods may not b abl to communicat (.g., du to tchnological barrirs). On attmpt to addrss this issu ar quasi unit disk graphs [19], whr ach vrtx is rprsntd by a pair o concntric disks, on o radius r, 0 < r < 1, and th othr o radius 1. In this modl, two vrtics ar connctd by an dg i thir radius-r disks ovrlap, and do not hav an dg i thir radius-1 disks do not ovrlap. Th rmaining dgs ar in or out o th graph on a cas by cas basis. In th wak unit disk modl such problms can b dalt with by simply dlting dgs btwn nods which ar narby but whos rangs do not ovrlap (.g., bcaus thy ar sparatd by a mountain rang). This givs us mor lxibility than quasi unit disk graphs. Formally, an dg-labling o a graph G = (V, E) is a map l : E {N, F }. I (u, v) E, thn (u, v) is calld nar i l(u, v) = N, and othrwis (u, v) is calld ar. In a unit disk (intrval) rprsntation I, ach vrtx v V is rprsntd as a disk (intrval) cntrd at th point I(v) R 2 (R). W dnot by I(u) I(v) th distanc btwn th points I(u) and I(v), and by a slight abus o notation, w also rr to I(v) as th disk (intrval) rprsnting v V. A wak unit disk (intrval) rprsntation o G with rspct to l is a rprsntation I such that or ach dg (u, v) E, I(u) I(v) t i and only i l(u, v) = N, or som ixd unit t > 0 (in othr words, th disks and intrvals hav diamtr t). Unlss othrwis statd, w assum t = 1. W say that a graph is wak unit disk (intrval) graphs i it has an appropriat rprsntation or all possibl dg-lablings. Rlatd Work: Wak unit ball graphs can b sn as a orm o graph drawing/labling whr a notion o closnss btwn vrtics is usd to din dgs, rom a givn st o prmissibl dgs. Thr ar many classs o graphs dind on som notion 2
o vrtx closnss. For xampl, proximity graphs ar thos that can b drawn in th plan such that vry pair o adjacnt vrtics satisis som ixd notion o closnss, whras vry pair o non-adjacnt vrtics satisy som notion o arnss [20]. Exampls o proximity graphs ar Gabril graphs, Dlaunay triangulations, and rlativ nighborhood graphs. Gabril graphs, dind in th contxt o catgorizing biological populations [13], can b mbddd in th plan so that or vry pair o vrtics (u, v), th disk with u and v as antipodal points contains no othr vrtx i and only i (u, v) is an dg. Rcntly, Evans t al. [10] studid rgion o inlunc graphs, whr ach pair o vrtics u, v in th plan is assignd a rgion R(u, v), and thr is an dg i and only i R(u, v) contains no vrtics, xcpt possibly u and v. Thy gnraliz this class o graphs to approximat proximity graphs, whr thr ar paramtrs ɛ 1 > 0 and ɛ 2 > 0, such that a vrtx othr than u or v is containd in R(u, v), scald by 1/(1 + ɛ 1 ), i and only i (u, v) is an dg; th rgion R(u, v), scald by 1 + ɛ 2, is mpty i and only i (u, v) is not an dg. Howvr thr is a signiicant dirnc btwn th notion o proximity graphs and th notion o wak unit ball graphs. In proximity graphs th notion o closnss is dind by two groups, namly adjacnt and non-adjacnt pairs o vrtics, whras or wak unit ball graphs, thr ar thr groups. Spciically, th nar and ar dgs in th input graph G rprsnt vrtx pairs with closnss and arnss rquirmnts, whil all nonadjacnt vrtx pairs in G hav no rquirmnt on proximity. Thus proximity graphs is mor rstrictd than th wak unit ball graphs, in that thy can b modld by wak unit ball graphs whr th input graph is th complt graph K n. Wak unit ball rprsntability in 1D is quivalnt to thrshold-coloring [1]. In this variant o graph coloring, intgr colors ar assignd to th vrtics so that ndpoints o nar dgs dir by lss than a givn thrshold, whil ndpoints o ar dgs dir by mor than th thrshold. Dciding whthr a graph is thrshold-colorabl with rspct to a givn partition o dgs into nar and ar is quivalnt to th graph sandwich problm or propr-intrval-rprsntability, which is known to b NP-hard [14]. Hnc, dciding whthr a graph admits a wak unit intrval rprsntation with rspct to a givn dg-labling is also NP-hard. Not that this is dirnt than rcognizing unit intrval graphs, which can b don in linar tim [11]. It is also known that planar graphs with girth (th lngth o a shortst cycl in th graph) at last 10 ar always thrshold-colorabl. Svral Archimdan lattics (which corrspond to tilings o th plan by rgular polygons), and som o thir duals, th Lavs lattics, ar thrsholdcolorabl [2]. Hnc, ths graph classs ar wak unit intrval graphs. Unit intrval graphs ar also rlatd to thrshold and dirnc graphs. In thrshold graphs thr xists a ral numbr S and or vry vrtx v thr is a ral wight a v so that (v, w) is an dg i and only i a v + a w S [21]. A graph is a dirnc graph i thr is a ral numbr S and or vry vrtx v thr is a ral wight a v so that a v < S and (v, w) is an dg i and only i a v a w S [16]. Not that or both ths classs th xistnc o an dg is compltly dtrmind by th thrshold S, whil in our stting th dgs dind by th thrshold (siz o th ball) must also blong to th original (not ncssarily complt) graph. Thrshold-colorability is also rlatd to th intgr distanc graph rprsntation [9, 12]. An intgr distanc graph is a graph with th st o intgrs as vrtx st and with an dg joining two vrtics u and v i and only i u v D, whr D is a subst o th positiv intgrs. 3
Our Rsults: W introduc th notion o wak unit disk and intrval rprsntations. Whil inding rprsntations with unit intrvals is quivalnt to thrshold-coloring whr som rsults ar alrady known, th problm o wak unit disk rprsntability is nw. W irst show that rcognizing wak unit disk graphs is NP-hard. Not that th NP-hardnss o th unit intrval variant ollows rom th rsults in [1]. W thn considr subclasss o planar graphs that admit wak unit disk (intrval) rprsntation. W show that vry dgr-2 contractibl graph (as dind latr) has a wak unit disk rprsntation. In particular, any sris-paralll graph (which includs all outrplanar graphs) has a wak unit disk rprsntation. For rprsntation with unit intrvals, it ollows rom [1] that all planar graphs with girth at last 10 ar wak unit intrval graphs. W gnraliz th rsult by proving that graphs o boundd maximum avrag dgr hav wak unit intrval rprsntations or any givn dg-labling. In th othr dirction, w construct an xampl o a planar girth-4 graph which is not a wak unit intrval graph, improving on th arlir girth-3 xampl. Furthr, w show that dns planar graphs do not always admit wak unit intrval graph rprsntation. Finally w study outrplanar graphs. It is known that som outrplanar graphs with girth 3 ar not wak unit intrval graphs, and our xampl o girth-4 graph is not outrplanar. Thus, a natural qustion in this contxt is whthr all girth-4 outrplanar graphs admit wak unit intrval rprsntation. W show that this is indd th cas. 2 Wak Unit Disk Graph Rprsntations First w considr th complxity o rcognizing wak unit disk graphs. Lmma 1. It is NP-hard to dcid i a graph G with an dg-labling l admits a wak unit disk rprsntation, vn i th dgs labld N induc a planar subgraph. Proo. It is known that dciding whthr a planar graph is a unit disk graph is NPhard [6]. Lt n b th numbr o vrtics o G, and din an dg-labling l o K n by stting l() = N i and only i is an dg o G. Clarly, a unit disk rprsntation o G is also a wak unit disk rprsntation o K n with rspct to l and vic vrsa. Not that Lmma 1 only provs NP-hardnss, and th problm o dciding whthr a graph with an dg-labling has a wak unit disk rprsntation is not known to b in NP. Th obvious approach is to us a wak unit disk rprsntation as a polynomial siz crtiicat. Unortunatly, it has rcntly bn showd that unit disks graphs on n vrtics may rquir 2 2Θ(n) bits or a unit disk rprsntation with intgr coordinats [22]. Unit Disk Rprsntation o Outrplanar and Rlatd Graphs Not that th class o wak unit disk graphs strictly contains th class o wak unit intrval graphs. For xampl, in Fig. 2, w provid a wak unit disk rprsntation o th sungraph, which dos not admit a wak unit intrval rprsntation. Our main goal hr is to prov that vry sris-paralll graph is a wak unit disk graph, or vry dglabling. To this nd, w study a largr class o graphs, calld dgr-2 contractibl graphs. A simpl graph G is a dgr-2 contractibl graph i on o th ollowing holds: 1. G is an indpndnt st; 4
u 0 v 1 u 1 u 0 v 1 u 1 v0 v 2 v 0 u 2 v 2 u 2 (a) (b) Fig. 2. (a) Th sungraph has no wak unit intrval rprsntation, but (b) it has a wak unit disk rprsntation. Nar/ar dgs ar indicatd with solid/dashd lin sgmnts. 2. G has an dg (u, v) such that v has dgr at most 2, and th graph obtaind by contracting (u, v) and rmoving paralll dgs is a dgr-2 contractibl graph. Not that dgr-2 contractibl graphs is a strict subclass o 2-dgnrat graphs. Thorm 1. Evry dgr-2 contractibl graph is a wak unit disk graph. Proo. W prov th thorm by induction on th numbr o dgs in a graph. Assum th inductiv hypothsis that vry dgr-2 contractibl graph G with m dgs has a wak unit disk rprsntation I with rspct to any dg-labling l so that (i) th disks hav diamtr t = 2 and (ii) or vry dg (x, y) o G, 1 < I(x) I(y) < 4. Th bas cas m = 0 is trivial, so assum that th claim holds or m > 0 and or any dgr-2 contractibl graph G with m < m dgs. Now considr an arbitrary dgr-2 contractibl graph G with m dgs and an arbitrary dg-labling l o G. I G has a vrtx o dgr 1, thn th dsird rprsntation can b constructd by rmoving th vrtx and considring a rprsntation or th rsulting graph. Hnc, w assum that G has no dgr-1 vrtics. Thn G has a vrtx v with xactly two nighbors u and w, such that contracting th dg (u, v) rsults in a dgr-2 contractibl graph G. W adopt th quivalnt convntion that, instad o contracting (u, v), w dlt v and add th dg (u, w) i it is not alrady prsnt. Clarly in G th numbr o dgs m < m. Thus by th inductiv hypothsis, G has a wak unit disk rprsntation I with rspct to th dg-labling l rstrictd to th dgs o G (i dg (u, w) dos not blong to G, giv it an arbitrary labl). Furthrmor, 1 < I (u) I (w) < 4. Without loss o gnrality, assum that I (u) = (0, 0) = p (say) and I (w) = (d, 0) = q (say), whr d > 0. Thn 1 < d < 4. W construct a rprsntation I o G by stting I(x) = I (x) or vry vrtx x v. To comput th valu o I(v), considr th ollowing our cass, basd on th valus o l(u, v) and l(w, v). Cas 1: l(u, v) = N, l(w, v) = N. I l(u, w) = N, i.., th disks or u and w intrsct ach othr, thn st I(v) to b th apx o an quilatral triangl with pq as a sid. Th disk or v thn intrsct both th disks or u and w. Othrwis, i l(u, w) = F, st I(v) = (0, d/2). Thn I(u) I(v) = I(w) I(v) = d/2 and sinc d < 4, d/2 < 2. Howvr sinc l(u, w) = F, w hav d > 2; hnc d/2 > 1. Cas 2: l(u, v) = N, l(w, v) = F. St I(v) to b (0, t), whr 1 < t < 2. Cas 3: l(u, v) = F, l(w, v) = N. St I(v) to b (0, t), whr d + 1 < t < d + 2. Cas 4: l(u, v) = F, l(w, v) = F. St I(v) to b th apx o an isoscls triangl with hight h and with pq as th bas, whr 2 < h < 4. Thn I(u) I(v) = I(w) I(v) = d = h 2 + (d/2) 2. Thus d > h > 2, so that th disks or u and w do not intrsct with th disk or v. Furthrmor sinc d < 4 and h < 4, d < 4. 5
Sris-paralll graphs ar dind as th graphs that do not hav K 4 as a minor [8]. Hnc by dinition, ths graphs ar closd undr dg contraction. It is also wllknown that a sris-paralll graph is subgraph o a 2-tr, which is dgr-2 contractibl, and that vry outrplanar graph is a subgraph o a sris paralll graph. Thus, by Thorm 1, w hav th ollowing corollary. Corollary 1. Evry outrplanar and sris-paralll graph is a wak unit disk graph. 3 Wak Unit Intrval Graph Rprsntations In this sction w study wak unit intrval rprsntability, which is quivalnt to thrshold graph coloring [1]. Givn a graph G = (V, E), an dg-labling l : E {N, F }, and intgrs r > 0, t 0, G is said to b (r, t)-thrshold-colorabl with rspct to l i thr xists a coloring c : V {1,..., r} such that or ach dg (u, v) E, c(u) c(v) t i and only i l(u, v) = N. Th coloring c is known as a thrshold-coloring. It is asy to s that thrshold-coloring is vry similar to wak unit intrval rprsntation, with th only dirnc that wak unit intrval graphs do not rquir intgr coordinats. Th nxt lmma shows that th dirnc is not signiicant. Lmma 2. A graph G has a wak unit intrval rprsntation or an dg-labling l i and only i G is (r, t)-thrshold-colorabl with rspct to l or intgrs r > 0, t 0. Proo. Clarly, a thrshold-coloring c is a wak unit intrval rprsntation o G with rspct to l (whr w us t as th unit o th rprsntation), so w nd only show that a wak unit intrval rprsntation o G is quivalnt to som thrshold-coloring. Suppos that I is a wak unit intrval rprsntation o G with rspct to l. I any o th intrvals o I intrsct only at thir ndpoints, thn w incras th lngth o ach intrval by som ɛ > 0, and choos ɛ so that th intrvals hav rational lngth. Nxt, w prturb th cntr point o ach intrval, in som ixd ordr, by som ɛ so that ach intrval is cntrd at a rational point. Nxt, w scal th rprsntation so that th cntr o ach intrval is an intgr, and th lngth o th intrvals is an intgr. Th modiid rprsntation is a thrshold-coloring (although r and t may b larg). Sinc dciding thrshold-colorability is NP-complt [1], so is th rcognition problm or wak unit intrval graphs. Lmma 3. It is NP-complt to dcid i a graph with an dg-labling admits a wak unit intrval rprsntation Nxt, w study wak unit intrval rprsntation or som graph classs. W irst prsnt a mthod or rprsnting graphs, which admit a dcomposition into a orst and a 2-indpndnt st. By G[U] w man th subgraph o G inducd by th vrtx st U V. Rcall that a subst I o vrtics in a graph G is calld indpndnt i G[I] has no dgs. Similarly, I is calld 2-indpndnt i th shortst path in G btwn any two vrtics o I has lngth gratr than 2. Such dcompositions hav bn applid to othr graph coloring problms [2, 3, 23]. Lmma 4. Suppos G = (I F, E) is a graph such that I is 2-indpndnt, G[F] is a orst, and I F =. Thn G is a wak unit intrval graph. 6
Proo. W assum that all intrvals in th proo ar cntrd at intgr coordinats and hav lngth t = 1. Suppos l : E {N, F } is an dg-labling. For ach v I, st I(v) = 0. Each vrtx in G[F] is assignd a point rom { 2, 1, 1, 2} as ollows. Choos a componnt T o G[F], and slct a root vrtx w o T. I w is ar rom a nighbor in I, st I(w) = 2; othrwis, I(w) = 1. Now prorm bradth irst sarch on T, assigning an intrval or ach vrtx as it is travrsd. Whn w rach a vrtx u w, it has on nighbor x in T which has bn procssd, and at most on nighbor v I. I v xists, w choos th intrval I(u) = 1 i l(u, v) = N, and I(u) = 2 othrwis. Thn, i th dg-labl (u, x) is not satisid, w multiply I(u) by 1. I v dos not xist, w choos I(u) = 1 or 1 to satisy th dg (u, x). By rpating th procdur on ach componnt o G[F], w construct a rprsntation o G. Rcall that th maximum avrag dgr o a graph G is th maximum o th avrag dgr o ach o its subgraphs H = (V H, E H ), and it is givn by mad(g) = max(2 E H / V H ), whr th maximum is takn ovr all subgraphs o G. It is known that vry planar graph G o maximum avrag dgr mad(g) strictly lss than 26 11 can b dcomposd into a 2-indpndnt st and a orst [7]. Hnc, Thorm 2. Evry planar graph G with mad(g) < 26 11 is a wak unit intrval graph. W also not that a planar graph with girth g satisis mad(g) < g 2 [4]. Thror, a planar graph with girth at last 13 has always a wak unit intrval rprsntation. Nxt w prsnt a gnralization o Lmma 4, suitabl or graphs which hav an indpndnt st that is in som sns narly 2-indpndnt. Th stratgy is to dlt crtain dgs so th indpndnt st bcoms 2-indpndnt, obtain a unit intrval rprsntation using Lmma 4, and thn modiy it so that it is a rprsntation o th original graph. Formally, lt I b an indpndnt st in a graph G. Suppos that or vry vrtx v I, thr is at most on vrtx u I such that th distanc btwn v and u in G is 2. Also suppos that thr is only on Fig. 3. Dcomposition o a graph path with two dgs conncting v to u. Thn w call into a narly 2-indpndnt st (rd I narly 2-indpndnt. Th pair {u, v} is calld an vrtics) and a orst (black vrtics I-pair, and th dgs o th path (u, x, v) conncting u and v ar calld I-dgs, which ar associatd and dgs). Thin blu ar I-dgs. with th I-pair {u, v}; s Fig. 3. Lmma 5. Lt G = (I F, E) b a graph, whr I is a narly 2-indpndnt st, G[F] is a orst and I F =. Thn G has a wak unit intrval rprsntation with rspct to any dg-labling l. Proo. Assum that all intrvals in th proo ar cntrd at intgr coordinats and hav siz t = 3. Suppos that l : E {N, F } is an dg-labling o G. Lt E E b a st such that or ach I-pair {u, v}, xactly on o th I-dgs associatd with {u, v} blongs to E. Lt G = (V, E E ). Thn clarly I is 2-indpndnt in G and G [F] is a orst; by Lmma 4, thr xists a wak unit disk rprsntation I o G or l. 2g 7
W now modiy I to construct a wak unit disk rprsntation I o G with rspct to l. First, or ach vrtx v V, st I(v) = 0 i I (v) = 0, I(v) = 2 i I (v) = 1, and I(v) = 5 i I (v) = 2 (i I (v) is ngativ, do th sam but st I(v) ngativ). It is clar that I is a wak unit disk rprsntation o G. Now, lt (x, y) E. On o ths vrtics, say x, is in I so I(x) = 0, and I(y) { 5, 2, 2, 5}. Without loss o gnrality assum that I(y) > 0; th cas whr I(y) < 0 is symmtric. Now it is possibl that l(x, y) = N but I(x) I(y) > 3 or that l(x, y) = F but I(x) I(y) 3. In th irst cas, w must hav I(y) = 5. W modiy I so that I(x) = 1 and I(y) = 4. Not that y is still nar to vrtics with intrvals cntrd at 2 or 5, and ar rom vrtics with intrvals cntrd at lss than 1. Similarly, x is still clos to th intrvals at 2, 0, or 2, but ar rom 5 and 5. Thus all th dgs o E E ar satisid by th modiication o I, and additionally th dg (x, y) is satisid. In th scond cas, w hav I(y) = 2. W modiy I so that I(x) = 1 and I(y) = 3. As bor, no dgs which disagrd with th dg-labling still disagr with th dg-labling. Sinc I is narly 2-indpndnt, our modiications to th rprsntation I will not act non-local vrtics, as vry vrtx in I is adjacnt to at most on dg o E. Wak Unit Intrval Rprsntation o Outrplanar Graphs It is known [1] that som outrplanar graphs containing triangls ar not wak unit intrval graphs,.g., th sungraph in Fig. 2. Hnc, w study wak unit intrval rprsntability o triangl-r outrplanar graphs. W start with a claim or girth 5. Lmma 6. An outrplanar graph with girth 5 is a wak unit intrval graph. Proo. W prov that girth-5 outrplanar graphs may b dcomposd into a orst and a 2-indpndnt st using induction on th numbr o intrnal acs. Th rsult will ollow rom Lmma 4. Th claim is trivial or a singl intrnal ac, so assum that it is tru or all girth-5 outrplanar graphs with k 1 intrnal acs. Lt G b a girth-5 outrplanar graph with k + 1 intrnal acs. Sinc G is outrplanar, it must hav at last on ac = (v 1,..., v l ), l 5, such that vry vrtx o xcpt v 1, v l is o dgr 2. Considr th graph G obtaind by dlting v 2,..., v l 1. Th vrtics o G hav a dcomposition into a 2-indpndnt st I and a st F such that G [F] is a orst. Now w will add th vrtics v 2,..., v l 1 to ithr I or F so that I is a 2-indpndnt st in G, and G[F] is a orst. I ithr o v 1, v l blongs to I, thn add all th rmaining vrtics to F. Othrwis, add v 3 to I and th rst to F. Sinc v 1, v l ar not in I, v 3 has distanc at last 3 rom any othr lmnt o I. Nxt our goal is to show that a triangl-r outrplanar graph G always has a wak unit intrval rprsntation or any dg-labling. W assum that all intrvals ar cntrd at intgr coordinats and w us intrvals o siz t = 2. Our stratgy is to ind a rprsntation o G by a travrsal in a dpth-irst sarch mannr o its wak dual graph G (th planar dual minus th outrac). W ind intrvals or all th vrtics in ach intrior ac o G as it is travrsd in G. Sinc w ar considring triangl-r graphs, this implis that w tak a path P n = (u 1, u 2,..., u n ), n 4, whr th two nd vrtics u 1 and u n ar alrady procssd and w nd to assign unit intrvals to th intrnal vrtics u 2,..., u n 1 o P n. W additionally maintain th invariant in our rprsntation that or ach dg (u, v) o G, I(u) I(v) 6. For a particular dg-labling 8
l o P n = (u 1,..., u n ), call a pair o coordinats x, y asibl i thr is a wak unit disk rprsntation I o P n or l with t = 2, whr I(u 1 ) = x, I(u n ) = y, and or any i {1,..., n 1}, I(u i ) I(u i+1 ) 6. W irst nd th ollowing thr claims. Claim 1 For any valu o x {2, 3, 2, 3}, th pair 0, x is asibl or any dglabling l o P 3 = (u 1, u 2, u 3 ). Proo. Without loss o gnrality, w may assum that x > 0. W comput a dsird wak unit disk rprsntation I with t = 2 or P 3 with rspct to l as ollows. Assign I(u 1 ) = 0 and I(u 3 ) = x. Assign I(u 2 ) in such a way that I(u 2 ) = 2 i l(u 1, u 2 ) = N, and I(u 2 ) = 3 i l(u 1, u 2 ) = F. Thn choos th sign o I(u 2 ) to b th sam as I(u 3 ) i l(u 2, u 3 ) = N, and th opposit o I(u 3 ) i l(u 2, u 3 ) = F. Claim 2 For any dg-labling o P 3 = (u 1, u 2, u 3 ), ithr 0, 4 or 0, 6 ar asibl. Proo. W comput a dsird wak unit disk rprsntation I with t = 2 or l as ollows. I l(u 1, u 2 ) = l(u 2, u 3 ) = N, thn I(u 1 ) = 0, I(u 2 ) = 2, and I(u 3 ) = 4. Othrwis, assign I(u 1 ) = 0, I(u 3 ) = 6, and I(u 2 ) = 2, 3 or 4 whn (l(u 1, u 2 ), l(u 2, u 3 )) hav valus (N, F ), (F, F ), and (F, N), rspctivly. Claim 3 For any intgr valu o x [ 6, 6], th pair 0, x is asibl or any dglabling o P n = (u 1, u 2,..., u n ), n 4. Proo. Without loss o gnrality, lt x 0. Considr irst th cas or n = 4. Tak a particular dg-labling l o P 4. For any intgr valu o 0 x 5, thr is at last on numbr y {2, 3, 2, 3} and at last on numbr z {2, 3, 2, 3} such that x y 2 and 2 < x z 6. In particular, it suics to choos or x = 0, y = 2, z = 3; or x = 1, 2, 3, 4, y = 2, z = 2 and or x = 5, y = 3, z = 2. Thus i 0 I(u 4 ) 5, and rgardlss o whthr l(u 3, u 4 ) is N or F, on can choos a valu or I(u 3 ) rom {2, 3, 2, 3} rspcting both th dg-labling o (u 3, u 4 ) and th proprty that I(u 3 ) I(u 4 ) 6. Thn by Claim 1, 0 and x is asibl or th dg-labling l o P 4. A similar argumnt shows that i l(u 3, u 4 ) = F, thn 0 and x = 6 is asibl. On th othr hand, i x = 6 and l(u 3, u 4 ) = N, thn both 4 and 6 ar valid choics or I(u 3 ). By Claim 2, 0 and 6 is asibl or any dg-labling l o P 4. Considr now th cas with n > 4. Thn assign coordinats I(u 1 ) = 0, I(u n ) = x and or i {n 1,..., 4}, assign I(u i ) [ 6, 6] such that it rspcts both l(u i, u i+1 ) and th proprty that I(u i ) I(u i+1 ) 6. Thn a similar argumnt as that or n = 4 can b usd to xtnd this rprsntation to u 2 and u 3. Th nxt corollary immdiatly ollows rom Claim 3. Corollary 2. Any pair x, y with x y 6, is asibl or any dg-labling o P n = (u 1, u 2,..., u n ), n 4. Thorm 3. Evry triangl-r outrplanar graph is a wak unit intrval graph. Proo. I G is not 2-connctd, w augmnt it in th ollowing way. Lt v b a cut vrtx o G and lt H 1,..., H k b th 2-connctd componnts o G containing v. For i {1,..., k 1}, lt u b a nighbor o v in H i, and w b a nighbor o v in 9
x v 10 v 11 v 2 v 3 v 1 v 1 w 2 v 0 u w 0 (a) v 2 (b) w 1 Fig. 4. (a) A whl graph W 11 with an dg-labling, that has no wak unit intrval rprsntation. (b) A girth-4 graph with an dg-labling, that has no wak unit intrval rprsntation. H i+1. Add th path (u, x, w), whr x is a nw vrtx. Clarly, any wak unit intrval rprsntation o th nw 2-connctd graph is also a wak unit intrval rprsntation o G, and th nw graph is outrplanar with girth 4. Now lt G b a 2-connctd triangl-r outrplanar graph with n > 4 vrtics mbddd in th plan with vry vrtx on th outrac, and lt l b an dg-labling o G. W nxt comput a wak unit intrval rprsntation o G or l. Th proo is by induction on th numbr o vrtics in G, with th n-vrtx cycl as a bas cas. Assum th inductiv hypothsis that vry triangl-r outrplanar graph with wr than n vrtics is a wak unit intrval graph. Furthr, assum that or such a graph G with any dg-labling l, thr is a wak unit intrval rprsntation o G or l whr any two nighbor vrtics u and v satisy I(u) I(v) 6. Clarly i G has at last two cycls, thn G has a path P k = (u 1,..., u k ), k 4 with dg(u i ) = 2 or som 1 < i < k. Th thorm ollows rom th inductiv hypothsis and Corollary 2. Planar Graphs without Wak Unit Intrval Rprsntations Planar graphs with high dg dnsity may not hav wak unit intrval rprsntations. First w prov th rsult or a whl graph, dind as W n, n 4, ormd by adding an dg rom a vrtx v 1 to vry vrtx o an (n 1)-cycl (v 2,..., v n, v 2 ). Lmma 7. A whl graph is not a wak unit intrval graph. Proo. Din an dg-labling l o W n by l(v 2, v n ) = F, l(v 1, v i ) = F or 3 i n 1, and vry othr dg labld N; s Fig. 4(a). Suppos I is a wak unit intrval rprsntation o W n with rspct to l. Sinc only on dg o th triangl (v 1, v 2, v n, v 1 ) is ar, I(v 1 ) I(v 2 ), hnc assum that I(v 1 ) < I(v 2 ). For 3 i n, i I(v i 1 ) > I(v 1 ), w hav I(v i ) > I(v 1 ), sinc l(v i 1, v i ) = N and ithr l(v 1, v i 1 ) or l(v 1, v i ) is F. Thn I(v 1 ) < I(v 2 ) I(v 1 ) + 1, and I(v 1 ) < I(v n ) I(v 1 ) + 1, contradicting that l(v 2, v n ) = F and I is wak intrval rprsntation. Using Lmma 7, it is asy to s that any maximal planar graph with V 4 is not a wak unit intrval graph. Indd, considr such a graph G = (V, E) and a vrtx v V ; th nighborhood N(v) = {u (v, u) E} togthr with v inducs a whl subgraph. Th obsrvation lads to th ollowing thorm. Thorm 4. Any planar graph G with mad(g) 11 2 is not a wak unit intrval graph. 10
Proo. To prov th claim, w show that a wak unit intrval planar graph has at most 11 V /4 6 dgs. Considr a vrtx v o a wak unit intrval planar graph G = (V, E) and assum it is mbddd in th plan. Th nighborhood o v is acyclic; othrwis v and its nighborhood induc a whl, which by Lmma 7 is not a wak unit intrval graph. Thus th numbr o dgs btwn any two nighbors o v is at most dg(v) 1, whr dg(v) is th dgr o v. Dnot th numbr or a vrtx v by s(v). Considr th sum, S = v s(v), takn ovr all vrtics o G. It is asy to s that S 2 E V. Lt T and T b th sts o triangular and non-triangular acs in an mbdding o G. For ach triangl x T ach o th dgs in x is countd onc in S. Thus, 2 E V 3 T T (2 E V )/3. Counting both sids o th dgs w gt 2 E 3 T + 4 T T + T (2 E + T )/4 (8 E V )/12, sinc T (2 E V )/3. Thus, rom Eulr s ormula V E + T + T = 2, w hav V E + (8 E V )/12 2 E 11 V /4 6. In [1] all xampls o graphs without thrshold-coloring (and thus, not wak unit intrval graphs) hav girth 3. W strngthn th bound by proving th ollowing. Lmma 8. Thr xist planar girth-4 graphs that ar not wak unit intrval graphs. Proo. Considr th graph in Fig. 4(b). Suppos thr xists a wak unit intrval rprsntation I. Without loss o gnrality suppos that I(w 2 ) > I(u). Lt us considr two cass. First, suppos I(v 2 ) < I(u). Sinc th dgs (u, v 2 ) and (u, w 2 ) ar labld F, it must b that I(v 2 ) < I(u) 1 and I(u)+1 < I(w 2 ). Thn vrtx x must b rprsntd by an intrval nar to both o ths, which is impossibl sinc I(v 2 ) I(w 2 ) > 2. Othrwis I(v 2 ) > I(u). Thn I(v 1 ) I(v 2 ) 1 > I(u), and I(u) < I(w 2 ) implis that I(v 1 ) < I(w 2 ). Similarly, I(w 1 ) < I(v 2 ). Now, ithr I(w 2 ) I(v 2 ), or I(v 2 ) < I(w 2 ). In th irst cas, w 2 is nar to v 1 sinc I(v 1 ) < I(w 2 ) I(v 2 ) and I(v 1 ) I(v 2 ) 1. Th scond cas lads to a similar contradiction. 4 Conclusion and Opn Problms In this papr w introducd th concpt o wak intrsction rprsntation o graphs and studid rprsntations o planar graphs with unit disks and unit intrvals. A natural utur dirction is to considr wak intrsction rprsntations or othr graph classs and/or with dirnt gomtric objcts. Nxt w list svral intrsting opn problms. 1. Dciding whthr a graph has a wak unit disk (intrval) rprsntation or a givn dg-labling is NP-hard. Howvr, th problm o dciding whthr a graph is a wak unit disk (intrval) graph is opn, and it rmains opn whn rstrictd to planar graphs. Not that th class o wak unit disk (intrval) planar graphs is not closd undr taking minors, as subdividing ach dg o a planar graph thr tims rsults in a planar graph with girth at last 10, which is a wak unit intrval graph. 2. Tightning th lowr and uppr bounds or maximum avrag dgr o wak unit intrval planar graphs, givn in Thorms 2 and 4, is a challnging opn problm. Basd on xtnsiv computr xprimnts, w conjctur that thr ar no wak unit intrval graphs with mor than 2 V 3 dgs. 11
3. W considrd planar graphs, but littl is known or gnral graphs. In particular, it would b intrsting to ind out whthr th dg dnsity o wak unit disk (intrval) graphs is always boundd by a constant. Rrncs 1. Alam, M.J., Chaplick, S., Fijavz, G., Kaumann, M., Kobourov, S.G., Pupyrv, S.: Thrsholdcoloring and unit-cub contact rprsntation o graphs. In: Graph-Thortic Concpts in Computr Scinc (WG 2013). LNCS, vol. 8165, pp. 26 37. Springr (2013) 2. Alam, M., Kobourov, S.G., Pupyrv, S., Toniskottr, J.: Happy dgs: Thrshold-coloring o rgular lattics. In: Fun with Algorithms, LNCS, vol. 8496, pp. 28 39. Springr (2014) 3. Albrtson, M.O., Chappll, G.G., Kirstad, H.A., Kündgn, A., Ramamurthi, R.: Coloring with no 2-colord P4. Elctron. J. Combin 11(1), R26 (2004) 4. Borodin, O., Kostochka, A., Nštřil, J., Raspaud, A., Sopna, E.: On th maximum avrag dgr and th orintd chromatic numbr o a graph. Dis. Math. 206(1), 77 89 (1999) 5. Brmnr, D., Evans, W., Frati, F., Hyr, L., Kobourov, S., Lnhart, W., Liotta, G., Rappaport, D., Whitsids, S.: On rprsnting graphs by touching cuboids. In: Graph Drawing. LNCS, vol. 7704, pp. 187 198. Springr (2012) 6. Bru, H., Kirkpatrick, D.G.: Unit disk graph rcognition is NP-hard. Computational Gomtry 9(1), 3 24 (1998) 7. Bu, Y., Cranston, D.W., Montassir, M., Raspaud, A., Wang, W.: Star coloring o spars graphs. Journal o Graph Thory 62(3), 201 219 (2009) 8. Duin, R.: Topology o sris-paralll ntworks. J. Math. Anal. Appl. 10, 303 318 (1965) 9. Egglton, R., Erdös, P., Skilton, D.: Colouring th ral lin. Journal o Combinatorial Thory, Sris B 39(1), 86 100 (1985) 10. Evans, W., Gansnr, E.R., Kaumann, M., Liotta, G., Mijr, H., Spillnr, A.: Approximat proximity drawings. In: Graph Drawing. LNCS, vol. 7704, pp. 166 178. Springr (2012) 11. d Fguirdo, C.M.H., Midanis, J., d Mllo, C.P.: A linar-tim algorithm or propr intrval graph rcognition. Inormation Procssing Lttr 56(3), 179 184 (1995) 12. Frrara, M., Kohayakawa, Y., Rödl, V.: Distanc graphs on th intgrs. Combinatorics Probability and Computing 14(1), 107 131 (2005) 13. Gabril, K.R., Sokal, R.R.: A nw statistical approach to gographic variation analysis. Systmatic Biology 18(3), 259 278 (1969) 14. Golumbic, M.C., Kaplan, H., Shamir, R.: Graph sandwich problms. Journal o Algorithms 19(3), 449 473 (1995) 15. Hal, W.K.: Frquncy assignmnt: Thory and applications. Procdings o th IEEE 68(12), 1497 1514 (1980) 16. Hammr, P.L., Pld, U.N., Sun, X.: Dirnc graphs. Dis. App. Math. 28(1), 35 44 (1990) 17. Hliněnỳ, P., Kratochvíl, J.: Rprsnting graphs by disks and balls (a survy o rcognitioncomplxity rsults). Discrt Mathmatics 229(1), 101 124 (2001) 18. Klist, L., Rahman, B.: Unit contact rprsntations o grid subgraphs with rgular polytops in 2D and 3D. In: Graph Drawing. pp. 137 148 (2014) 19. Kuhn, F., Wattnhor, R., Zollingr, A.: Ad hoc ntworks byond unit disk graphs. Wirl. Ntw. 14(5), 715 729 (2008) 20. Liotta, G.: Proximity drawings. In: Tamassia, R. (d.) Handbook o Graph Drawing and Visualization. Chapman & Hall/CRC (2007) 21. Mahadv, N.V., Pld, U.N.: Thrshold Graphs and Rlatd Topics. North Holland (1995) 22. McDiarmid, C., Müllr, T.: Intgr ralizations o disk and sgmnt graphs. Journal o Combinatorial Thory, Sris B 103(1), 114 143 (2013) 23. Timmons, C.: Star coloring high girth planar graphs. Th Elctronic Journal o Combinatorics 15(1), R124 (2008) 12
Appndix Acknowldgmnts W thank Michalis Bkos, Gaspr Fijavz, and Michal Kaumann or productiv discussions about svral variants o ths problms. A Unit-Cub Contact Rprsntations o Planar Graphs Thrshold coloring [1] is motivatd by th gomtric problm o unit-cub contact rprsntation o planar graphs [5]. In such a rprsntation, ach vrtx o a graph is rprsntd by a unit-siz cub and ach dg is ralizd by a common boundary with non-zro ara btwn th two corrsponding cubs. Th problms ar rlatd, as thrshold-coloring can b usd to ind unit-cub contact rprsntations. In particular i a graph G has a unit-squar contact rprsntation Γ in th plan and i H is any subgraph o G ormd by th nar dgs o G in som thrshold-coloring, thn on can ind a unit- cub-contact rprsntation o H. Formally, lt H b a thrshold subgraph o G = (V, E) and lt c : V {1... r} b an (r, t)-thrshold-coloring o G with rspct to an dg-labling l. W now comput a unit-cub contact rprsntation o H rom Γ using c. Assum (atr possibl rotation and translation) that th bottom ac or ach cub in Γ is co-planar with th plan z = 0; s Fig. 5(a). Also assum (atr possibl scaling) that ach cub in Γ has sid lngth t +ɛ, whr 0 < ɛ < 1. Thn w can obtain a unit-cub contact rprsntation o H rom Γ by liting th cub or ach vrtx v by an amount c(v) so that its bottom ac is at z = c(v); s Fig. 5(b). Not that or any dg (u, v) with l(u, v) = N, th rlativ distanc btwn th bottom acs o th cubs or u and v is c(u) c(v) t < (t+ɛ); thus th two cubs maintain contact. On th othr hand, or ach pair o vrtics u, v with l(u, v) N, on o th ollowing two cass occurs: (i) ithr (u, v) / E and thir corrsponding cubs rmain non-adjacnt as thy wr in Γ ; or (ii) l(u, v) = F and th rlativ distanc btwn th bottom acs o th two cubs is c(u) c(v) (t+1) > (t + ɛ), making thm non-adjacnt. d b a c b d a c d 1 b 2 3 a 1 1 c 0 b d a 21 1 c 3 1 (a) (b) Fig. 5. (a) A graph G with a unit-squar contact rprsntation. (b) A thrshold-coloring o G with thrshold 1 and a unit-cub-contact rprsntation o th subgraph ormd by nar dgs [1]. Th vrtx colors corrspond to th lvation o th cubs, whil th thrshold givs th siz o th cubs. In th igur w indicat N dgs with solid lins and F dgs with dashd lins. 13