c-domiatig Sets for Families of Graphs Kelsie Syder Mathematics Uiversity of Mary Washigto April 6, 011 1
Abstract The topic of domiatio i graphs has a rich history, begiig with chess ethusiasts i the 1850s determiig how may quees are ecessary to domiate a etire chessboard ad cotiuig to curret problems ivolvig computer commuicatio etworks, social etwork theory, ad other similar problems. We defie a domiatig set of a graph G to be a set of vertices of G such that every vertex of G is either i the set or adjacet to a vertex i the set. The domiatio umber for a graph G is the size of a miimum domiatig set. Determiig the domiatio umber of graphs ca prove highly useful i solvig may types of problems, ad recet studies of domiatig sets reflect this. We focus o describig various families of graphs i terms of bouds o the domiatio umber. Although the computatio of domiatig sets for arbitrary graphs is a NP-complete problem, it is possible to compute certai bouds o the domiatio umber for certai families of graphs. We examie families of graphs, specifically the family of grids, ad determie the bouds o domiatio umber for these families. We compare the domiatio umbers for the various classes of grids with other commo families of graphs.
1 Itroductio to Domiatio Our work focuses o the cocept of domiatio i graphs, a topic that has bee studied extesively i recet decades. We begi with a overview of the relevat defiitios alog with a series of examples. First, we review a few graph theory basics that are relevat throughout our work. More o the basics of graph theory ca be foud i [3], for istace. We defie a graph G = (V, E) as a pair of sets, with V the oempty set of vertices of G, ad E the set of edges betwee distict vertices of G. The cardiality of V, deoted V, represets the umber of vertices i G, while vertices u ad v are adjacet if the edge uv is i E. Throughout our work we cosider oly udirected simple graphs G = (V, E), that is, graphs with o directio o the edges ad at most oe edge betwee each pair of vertices. We defie the eighborhood of a vertex v (also called the closed eighborhood of v), as the set of vertices cosistig of v ad each vertex adjacet to v: N[v] = {v} {x V : vx E}. The degree of a vertex v, writte d(v), is defied as the umber of edges icidet with v. We deote the miimum degree of G with δ(g). Also, we defie d(u, v) as the distace betwee two vertices u ad v i V. A subset S of V is a domiatig set if every vertex v V is domiated by some elemet of S, that is, every v V is either a elemet of S or is adjacet to a elemet of S. The domiatio umber, γ(g) is the miimum cardiality of a domiatig set S of V. Fially, the graph G is c-domiated if for 0 < c 1, γ(g) c V. We use the cocept of c- domiatio throughout our work to describe the domiatio umbers for the graphs from various families of graphs. 1.1 Applicatios of Domiatio i Graphs The study of domiatio i graphs has historical roots as early as the 1850s whe Europea chess ethusiasts studied the problem of domiatig quees, as described i []. These ethusiasts worked to determie the miimum umber of quees ecessary so that every square o a stadard 8 8 chessboard is either occupied by a quee or ca be directly attacked by a quee, that is, every square is domiated by a quee. It was determied that a miimum of five quees are eeded, ad this problem ca be modeled by fidig a domiatig set of five quees. The mathematical study of domiatig sets bega i earest i the 1960s, ad sice that time, domiatig sets have bee used for may differet applicatios. Oe of the families of graphs we examie i detail is the family of graphs resemblig grids, which we defie i more detail i Sectio. A practical applicatio demostratig the importace of domiatio umbers i these grids utilizes grids to model city blocks. We let the vertices of the grid represet street corers ad the edges represet streets betwee corers. The, for example, cosider the problem of determiig how may police officers should be statioed across the city o street corers so that every corer is visible to at least oe officer. We assume that each officer ca view the corer o which they are statioed ad each corer that is o more tha oe street block away. The domiatio umber of the grid represetig the city would provide the miimum umber of police officers ecessary so that each corer is visible to at least oe officer. Fidig a miimum domiatig set would provide a descriptio of the street corers where officers should be statioed i order to accomplish this. Clearly may
other factors would eed to be cosidered if a city were tryig to solve such a problem, but domiatig sets could serve as a importat tool for decisio makers. Domiatig sets ca be used to model may other problems, icludig may relatig to computer commuicatio etworks, social etwork theory, lad surveyig, ad other similar issues. Determiig the domiatio umber for graphs ad fidig miimum domiatig sets could thus prove very useful. However, fidig the domiatio umber of a geeral graph G is a NP-complete problem []. Our work focuses o examiig certai families of graphs ad fidig the exact domiatio umbers for the graphs from these families. 1. Domiatio i Commo Families of Graphs We start by cosiderig a few examples of domiatio i commo families of graphs. Throughout our work, we focus o families of graphs with relatively small bouds o the domiatio umber. For example, cosider the family of complete graphs. A complete graph K o vertices is a graph i which each vertex is adjacet to every other vertex, that is, for every vertex v V, d(v) = 1. We see by ispectio that for all complete graphs, γ(k ) = 1, sice each vertex v V domiates itself ad its 1 eighbors. Thus the set D = {v} domiates the etire graph. Hece, we see that for a complete graph K, γ(k ) = 1 V = 1, ad so we say K is 1 -domiated. I Figure 1, we see a complete graph o 6 vertices with a domiatio umber of 1. I this figure, as well as the remaiig figures throughout our work, we use grey vertices to represet the vertices i the domiatig set D. u K 6 W 5 K,3 v Figure 1: Miimum domiatig sets for the graphs K 6, W 5 ad K,3 Aother family of graphs with a costat domiatio umber of 1 is the family of wheel graphs. A wheel graph W is a graph with vertices that cotais a cycle of legth 1 ad a cetral vertex v that is ot i the cycle but that is adjacet to every vertex i the cycle. Sice v is adjacet to every other vertex i W, it is clear that v domiates the etire graph. Thus, for all, γ(w ) = 1 V = 1, ad so W is also 1 -domiated. We see the wheel graph W 5 i Figure 1. Next, cosider the family of complete bipartite graphs, that is, graphs i which the vertices ca be partitioed ito two disjoit subsets U ad V, so that each edge coects a vertex from U to V, ad every vertex i U is adjacet to every vertex i V. For all complete bipartite graphs K,m with, m (where U cotais vertices ad V cotais m vertices), we see by ispectio that γ(k,m ) =, sice ay vertex u U domiates each of its m eighbors i V ad ay vertex v V domiates its eighbors i U. We have
the that K,m is -domiated, sice γ(k +m,m) = V =. For example, cosider the +m complete bipartite graph K,3 i Figure 1. We see that i these three families of graphs, the domiatio umber is costat i relatio to. I cotrast, for a geeral graph G with miimum degree 1 o vertices, it has bee prove that G is 1 -domiated [6], so that the boud o the domiatio umber of G icreases liearly with. For the families of graphs that we examie throughout the remaider of our work, the domiatio umber will also icrease with, though we will prove that the boud o the domiatio umber is sigificatly less tha the geeral boud for graphs with miimum degree 1. 1.3 Domiatio i Graphs with Miimum Degree Two The domiatio umber for graphs with miimum degree two has bee explored by William McCuaig ad Bruce Shepherd ad prove i [5]. The result provides a boud o the domiatio umber of such graphs with oly seve exceptios (details o the exceptioal graphs ca be foud i the origial paper [5]). Theorem 1.1. (McGuaig ad Shepard [5]) If G = (V, E) is a coected graph with miimum degree greater tha or equal to (δ(g) ) ad G is ot a graph of type B of exceptioal graphs, the γ(g) V. 5 Thus with this result, we see that if a graph G is coected ad cotais o vertices of degree 1, ad is ot oe of the seve bad graphs i B, the boud o domiatio umber is V. 5 Domiatio i Grids We cosider the family of grid graphs, a family of graphs with miimum degree but which has bouds o domiatio umber lower tha the geeral boud give i Theorem 1.1..1 Defiitio of Grids For the remaider of our work, we look at the class of grid graphs, ad we determie bouds o the domiatio umber based o the size of the grid. Such graphs resemble two-dimesioal grids ad ca be used to model thigs as importat as city blocks, ad therefore could be used i applied problems related to city cogestio ad/or traversal of streets. I formal terms, a two-dimesioal grid graph is a m graph G(m ) that is the graph Cartesia product of two paths of legth m ad, respectively. Rather tha explaiig the techical defiitio of the graph Cartesia product, we explai through a example. Figure illustrates the 3 5 grid G(3 5) o 15 vertices, which is the graph Cartesia product of the path of legth 3 ad the path of legth 5, respectively. For the purposes of this paper, we defie several classes of vertices i relatio to grids. Let a corer vertex be defied as oe of the four vertices of degree two that occurs i the first or the -th colums of a grid. We defie a outside vertex as oe of the vertices of
Figure : The grid G(3 5) degree three that occurs i the first or the -th colums of a grid, or i the first or m-th rows of the grid. We defie a iside vertex as oe of the vertices of degree four that occurs i the secod through ( 1)-st colums or the secod through (m 1)-st rows of the grid.. The Case of G( ) Cosider grids of size o vertices, that is, all grids with two rows ad colums (or, equivaletly, rows ad two colums). We will prove that the graph G( ) has domiatio umber + 1 γ(g( )) =. Our techique is to show that the expressio provides both a upper ad a lower boud. The upper boud argumet is costructive, while the lower boud will require a more techical proof. Lemma.1. The graph G( ) has domiatio umber satisfyig + 1 γ(g( )). Proof. We give a explicit costructio of a set of vertices that domiate the graph G( ) ad meet the boud. We break ito cases depedig o whether is eve or odd. Case 1: is eve. Let D be a subset of V such that D cotais oe vertex i each odd-umbered colum k, alteratig betwee the first ad secod rows, ad oe vertex i the -th colum, as see i Figure 3. That is, if D cotais the vertex i the first colum i the secod row, the D also cotais the vertex i the third colum i the first row, ad the vertex i the fifth colum i the secod row, ad so forth, as well as oe vertex i the -th colum i either row. The, D cotais exactly + 1 = vertices. Let D1 be the subset of D cotaiig all vertices of D except the vertex i the -th colum, so that D 1 =. For each pair of vertices u ad v i D 1, d(u, v) >, so that N[u] N[v] =, ad each vertex i V is adjacet to o more tha oe vertex i D 1. The subset D 1 cotais oe corer vertex i G that domiates itself ad its two eighbors, while all other vertices i D 1 are of degree three ad domiate themselves ad their three eighbors. So, D 1 domiates ( ) 1 + (1 3) = + 3 = 1
vertices i G. The, D 1 domiates the first 1 colums of G, which cotai ( 1) = vertices, as well as oe of the vertices i the -th colum of G. Now, let w be the vertex i D i the -th colum. We see that w domiates both itself ad the other vertex i the -th colum, so that D = D 1 {w} domiates G. k = 1 k = 5 k = 9 k = 3 k = 7 k = 11 k = 1 k = 5 k = 9 k = 1 k = 3 k = 7 k = 11 G( 11) G( 1) Figure 3: Miimum domiatig sets for the grids G( 11) ad G( 1) Case : is odd. Let D be a subset of V such that D cotais oe vertex i each odd-umbered colum k, alteratig betwee the first ad secod rows, as above ad as see i Figure 3. The, D cotais exactly vertices. Note that whe is odd, =. Agai, for each pair of vertices u ad v i D, d(u, v) >, so that N[u] N[v] =, ad each vertex i V is adjacet to o more tha oe vertex i D. Sice D cotais two corer vertices i G ad all other vertices i D are of degree three, D domiates ( ) + 1 + ( 3) = ( + 1) 8 + 6 = + 8 + 6 = vertices. But G cotais exactly vertices, ad so D domiates G. By the case aalysis above, it follows that γ(g( )). Lemma.. The graph G( ) has domiatio umber satisfyig + 1 γ(g( )). Proof. We will prove that o set with less tha vertices ca domiate G. Suppose to the cotrary that D is a miimum domiatig set with fewer tha vertices. Case 1: is odd. We are assumig that D <, ad sice is odd, it follows that D 1 = 1. All vertices i G have degree of either two or three, so that each vertex i D domiates at
most itself ad three adjacet vertices. Hece, the umber of domiated vertices, dom(d) satisfies dom(d) 1 ( ) 1 + 3 = 1 + 3 3 = =. But G( ) has vertices, ad so D caot domiate G. Case : is eve. We are assumig that D <, ad sice is eve, it follows that D. All vertices i G have degree of either two or three, so that each vertex i D domiates at most itself ad three adjacet vertices. Hece, the umber of domiated vertices, dom(d) satisfies dom(d) ( ) + 3 = + 3 = =. This boud is met if ad oly if each vertex i D is of degree three ad the eighborhoods of the vertices i D are all disjoit. However, if D cotais oly vertices of degree three, the i order to domiate all four corers, both vertices i the d colum ad both vertices i the ( 1)-st colum must be i D, cotradictig the disjoit eighborhoods of vertices i D. Thus, D caot domiate G with D <. By the case aalysis above, it follows that γ(g( )). Theorem.3. The graph G( ) has domiatio umber satisfyig + 1 γ(g( )) =. Proof. Follows immediately from Lemmas.1 ad.. Thus, the domiatio umber of grids G( ) is approximately = V, which is sigificatly less tha the boud prove for geeral graphs with miimum degree. We will see that this is true eve with larger grids G(3 ) ad G( ). As we cosider grids G(m ) with m >, we quickly see that provig bouds o the domiatio umber of these grids becomes far more complicated tha i the case of G( ). I fact, we will oly provide bouds for the cases of G(3 ) ad G( ). These bouds ad the techiques used i provig them could potetially be used to provide bouds for grids G(m ) with m 5, but we will ot examie these cases i detail..3 The Case of G(3 ) We cosider grids of the form 3 o 3 vertices ad prove that graphs of this form have domiatio umber satisfyig 3 + 1 γ(g(3 )) =.
We use the same techique as with grids, first providig a costructio of a domiatig set for G(3 ), ad the a more techical proof usig strog iductio o to prove that o set with fewer tha 3 vertices ca domiate G(3 ). Lemma.. The graph G(3 ) has domiatio umber satisfyig 3 + 1 γ(g(3 )). Proof. We will give a explicit costructio of a set of vertices that domiates the graph G(3 ) ad meets the boud. Figure shows such a costructio for grids G(3 1) through G(3 9). Figure : Miimum domiatig sets for the grids G(3 1) through G(3 9) For grids with 5 the costructios i Figure suffice. As we cosider a geeral costructio for grids with > 5, we refer to the cofiguratio of vertices i D i the costructio of the domiatig set for G(3 5), as see i Figure. The idea for larger grids is to repeat this patter. By lookig at the various cogruece classes for modulo, it is ot difficult to obtai the lower boud of the lemma. Lemma.5. The graph G(3 ) has domiatio umber satisfyig 3 + 1 γ(g(3 )). Proof. The proof techique is strog mathematical iductio, followed by a careful case aalysis. We oly provide a overview of the proof, ad refer the reader to [7] for the details of the argumet. The strog mathematical iductio is o, the legth of the grid. That is, we use iductio o to show that γ(g(3 )) 3. First, we ote that 3 = 1. Base case: = 1. We see by ispectio that γ(g(3 1)) = 1 = base case. 3(1)+1, so that the boud holds for the
Iductive step: Fix a ad suppose that for all values k less tha or equal to, γ(g(3 )) 3 = 1. We will prove that γ(g(3 ( + 1))) 3()+1 = 3+ = + 1. Let G = G(3 ( + 1)), ad let D be a domiatig set of G. The D must domiate a, b, ad c, the vertices i the ( + 1)-st colum, i additio to the remaiig vertices i G. We ow cosider the followig cases as we cosider how a, b, ad c ca be domiated. Case 1: a, b, c D. I order for D to domiate a, b, ad c, we have that d, e, f D. Cosider G = (V, E ) = G\{a, b, c, d, e, f, g, h, i}, that is, cosider the graph G, obtaied by removig the three rightmost colums from G, so that G = G(3 ( )), as see i Figure 5. Now we cosider a set D that domiates G. Let D be D V plus all vertices u i the ( )-d colum of G such that u is domiated by some vertex v i the ( 1)-st colum. If there are m such vertices u, we have that D D 3 m + m = D 3. The, D domiates G = G(3 ( )). By the iductive hypothesis, we kow that D ( ) 3. Furthermore, we have that D D + 3 + 1 3 + 1. Thus, the boud holds i this case. m j g d a k h e b o l i f c } {{ } G Figure 5: A G(3 ( + 1)) grid with a, b, c D Case : At least two of a, b, c are i D. I this case, all three vertices are domiated by D. Deletig the last two colums of G as i Case 1, we obtai a similar coclusio. Case 3: Exactly oe of a, b, c is i D. This is the most difficult case i that we must cosider two possibilities, either a D (or c D) or b D, ad each will require multiple subcases. However, the argumet for each case is very similar to the argumets give i Cases 1 ad above, ad we omit the details. So, by strog iductio o, we have thus show that γ(g(3 )) = 1 = 3. Theorem.6. The graph G(3 ) has domiatio umber satisfyig 3 + 1 γ(g(3 )) =. Proof. Follows immediately from Lemmas. ad.5.
We see the that G(3 ), a graph o 3 vertices, is domiated by 3 vertices, so that G(3 ) is approximately 1 -domiated. As with the families of graphs we have previously examied, this is sigificatly lower tha the boud provided i Theorem 1.1. 5. The Case of G( ) Before cosiderig the geeral case of G( ), we cosider the grid G( ). Two equivalet patters of vertices i a miimum domiatig set for G( ) are show i Figure 6. a b Figure 6: Equivalet miimum domiatig sets for the grid G( ) Provig that these sets domiate G( ) is straight-forward. Coutig ca the be used to argue that o smaller sets ca domiate ad that these sets are, i fact, the oly miimum domiatig sets. We state this precisely i the followig theorem. Theorem.7. The graph G( ) has domiatio umber γ(g( )) =. Moreover, domiatig sets give i Figure 6 are the oly sets of size four that domiate G( ). Miimum domiatig sets ca be computed similarly for grids G( ) with 10. We used the software package Magma [1] to calculate the domiatio umber for these grids, ad cofiguratios of a miimum domiatig set for each of these grids is show i Figure 7. We provide the followig proof of the geeral case where 10, usig the cofiguratio of vertices i the grid G( ) as a basis for the geeral costructio. Figure 7: Miimum domiatig sets for the grids G( 1) through G( 10)
For all 10, we ca provide a explicit costructio of a set of vertices D that domiates the graph G = G( ) ad meets the boud γ(g( )). The idea is quite simply to repeat the patter give by a miimum domiatig set for G( ) as show i Theorem.7. By cosiderig cases o the cogruece class of modulo, oe ca easily fid patters of vertices that domiate the grids. To show that γ(g( )), we start with a miimal couter-example. That is, we cosider the miimum value k so that γ(g( k)) < k. After a careful case aalysis, this leads to a cotradictio, givig us our mai result. Theorem.8. For 10, the graph G( ) has domiatio umber satisfyig γ(g( )) =. Sice G( ) is a graph o vertices that is domiated by vertices, we see that G( ) is 1 -domiated. Thus, as with the case of G( ) ad G(3 ), we see that the domiatio umber for grids G( ) is sigificatly less tha the geeral boud provided by Theorem 1.1. 3 Coclusio We have idetified several commo families of graphs with domiatio umbers sigificatly lower tha the boud γ(g) V for geeral graphs G with miimum degree. Though we 5 did ot examie larger graphs of size G(m ) with m 5, the techiques we used to fid the bouds for the smaller sizes of grids could be applied to fid at least a upper boud for the domiatio umber of larger grids. For example, for a grid of size G(8 ), the patter used to provide a costructio for a miimum domiatig set of G( ) could be repeated to create a costructio of a miimum domiatig set cotaiig vertices, thus provig that γ(g(8 )). However, we will ot examie these larger grids further. Table 1: Domiatio umbers of family of graphs Family Notatio Domiatio Number c Graphs with δ(g) 1 G γ(g) V 1 Coected Graphs with δ(g) G γ(g) V 5 5 Complete Graphs ( 3) K γ(k ) = 1 1 1 Wheel Graphs ( ) W γ(w ) = 1 Complete Bipartite Graphs (, m ) K,m γ(k,m ) = Grids G( ) γ(g( )) = 3 Grids G(3 ) γ(g(3 )) = 3 1 Grids ( 10) G( ) γ(g( )) = +m 1 1 Table 1 summarizes the differet families of graphs we have examied, the domiatio umber associated with each family, ad the value of c associated with each family idicatig that the family is c-domiated.
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