Criticality. of the lower domination. parameters. of graphs

Transcription

1 Criticality of the lower domination parameters of graphs Audrey Coetzer Thesis presented in partial fulfilment of the requirements for the degree Master of Science in Applied Mathematics at the Department of Mathematical Sciences of the University of Stellenbosch, South Africa. Supervisor: Dr PJP Grobler March 2007

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3 Declaration I, the undersigned, hereby declare that the work contained in this thesis is my own original work and that I have not previously in its entirety or in part submitted it at any university for a degree. Signature: Date: i

4 Abstract In this thesis we focus on the lower domination parameters of a graph G, denoted π(g), for π {i, ir, γ}. For each of these parameters, we are interested in characterizing the structure of graphs that are critical when faced with small changes such as vertex-removal, edge-addition and edge-removal. While criticality with respect to independence and domination have been well documented in the literature, many open questions still remain with regards to irredundance. In this thesis we answer some of these questions. First we describe the relationship between transitivity and criticality. This knowledge we then use to determine under which conditions certain classes of graphs are critical. Each of the chosen classes of graphs will provide specific examples of different types of criticality. We also formulate necessary conditions for graphs to be ir-critical and ir-edge-critical. ii

5 Opsomming In hierdie tesis fokus ons op die onderste dominasieparameters van n grafiek G, genaamd π(g), vir π {i, ir, γ}. Vir elkeen van hierdie parameters stel ons belang in die karakterisering van die struktuur van grafieke wat krities is met betrekking tot klein veranderinge soos die verwydering van n nodus of n lyn, of die byvoeging van n lyn. Terwyl daar al vele resultate in die literatuur is oor die kritiekheid van onafhanklikheid en dominasie, bestaan daar nog heelwat oop vrae oor onoorbodigheid. In hierdie tesis poog ons om van die vrae te beantwoord. Ons beskryf eers die verhouding tussen transitiwiteit en kritiekheid. Met hierdie kennis bepaal ons dan die voorwaardes waaronder sekere klasse van grafieke krities is. Elkeen van die gekose klasse verskaf vir ons spesifieke voorbeelde van die verskillende tipes kritiekheid. Ons formuleer ook noodsaaklike voorwaardes waaronder n grafiek krities is met betrekking tot onoorbodigheid as n nodus verwyder word of n lyn bygevoeg word. iii

6 Acknowledgements The author hereby wishes to express her gratitude toward: The financial assistance of the post graduate bursary office and the Department of Applied Mathematics of the University of Stellenbosch. Dr PJP Grobler for his guidance, patience and mentoring during the course of this degree. Prof J van Vuuren for the use of his students facilities and for inspiring her towards the field of Graph Theory. Her family and friends for their support and encouragement. iv

7 Glossary Adjacent: Two vertices of a graph G are said to be adjacent if there exists an edge of G joining the two vertices. Annihilate: For a given graph G, if the private neighbourhood of s S relative to S for S V G is completely contained in the closed neighbourhood of v V G S, then we say v annihilates s relative to S. Automorphism: An automorphism of a graph G is an isomorphism of G onto itself. Circulant: The circulant graph G = C n a 1,a 2,,a l is a graph with 0 < a 1 < a 2 < < a l < n, vertex set V G = {v 1,v 2,,v n } and edge set E G = {{v i,v i+j } and {v i,v i j } : i = 1, 2,,n and j = a 1,a 2,,a l }. Closed Neighbourhood: The closed neighbourhood of a vertex v in a graph G is the set of all vertices adjacent to v in G, as well as v itself, and is denoted N[v]. The closed neighbourhood of a vertex set S in G is defined as N[S] = {N[v] : v S}. Complement: The complement G of a graph G is the graph for which V G = V G and e E G if and only if e E G. Complete Graph: A complete graph of order n, denoted by K n, is a graph in which every pair of vertices are adjacent. Complete Multipartite Graph: The complete multipartite graph K n1,n 2,,n m is the complement of the disjoint union of complete graphs K n1 K n2 K nm. v

8 Component: A subgraph H of a graph G is called a component of G if H is a maximally connected subgraph of G. Connected: For vertices u and v of a graph G, u is said to be connected to v if G contains a u v path. The graph G is called a connected graph if the vertices u and v are connected for any pair u,v V G. Copy: A copy of G is a graph isomorphic to G. Degree: The degree of a vertex v of a graph G is the cardinality of the open neighbourhood of v in G, and is denoted deg G v. Disconnected: A graph that is not connected is said to be disconnected. Disjoint: Two graphs G and H are disjoint if V G V H =. Domination Number: The (lower) domination number, denoted γ(g), of a graph G is the minimum size over all minimal dominating sets of G. Dominating Set: A vertex subset S V G of G is called a dominating set if every vertex v V G S is adjacent to a vertex u S. Edge: An edge is a 2 element subset of the vertex set of a graph. Edges are indicated by inter connecting lines between vertices in graphical representations of a graph. Edge Set: The set E G, comprised of all the edges of a graph G, is called the edge set of the graph. Edge-transitive: G is edge-transitive if for any {u 1,u 2 }, {v 1,v 2 } E G there exists an automorphism Φ of G such that Φ({u 1,u 2 }) = {v 1,v 2 }. External Private Neighbourhood: For a vertex subset S of a graph G, a vertex w V G S is called an external private neighbour of v relative to S, if N(w) S = {v}. The set of all epns of v is called the external private neighbourhood of v relative to S, and is denoted epn(v,s). vi

9 Graph: A graph is a finite, nonempty set of elements, called vertices, together with a (possibly empty) set of 2 element subsets of the vertex set called edges. A graph may be represented graphically as a set of nodes with inter connecting lines. Independence Number: The maximum cardinality over all maximal independent sets of a graph G is called the independence number of G and is denoted β(g). Independent Domination Number: Any dominating set of a graph G that is also independent is called an independent dominating set of G, the minimum cardinality of which is called the independent domination number, denoted i(g). Independent Set: A vertex subset S of a graph G is called independent if no two vertices in S are adjacent in G. Induced Subgraph: For a non empty subset S V G of a graph G the so called induced subgraph of S in G, denoted S G, is the subgraph of G with vertex set V S G = S and edge set E S G = {uv E G : u,v S}. Irredundance Number: The irredundance number, denoted IR(G), is the largest number of vertices in a maximal irredundant set of G. Irredundant: For S V G and s S, s is a irredundant vertex of S if s is not redundant. Irredundant Set: S V G is an irredundant set of G if the private neighbourhood of each vertex s S relative to the set S is empty. Isolated Vertex: A vertex in graph G is isolated if it is adjacent to no other vertices of G. Isomorphism: An one to one mapping φ : V G V H between the vertex sets of two graphs G and H such that uv E G if and only if φ(u)φ(v) E H. Isomorphic: Two graphs G and H are called isomorphic, written as G = H, if there exists an isomorphism between their two vertex-sets. vii

10 Lower Independence Number: The lower independence number i(g) is the smallest number of vertices in a maximal independent set of G. Lower Irredundance Number: The irredundance number IR(G) and the lower irredundance number ir(g) are the largest and smallest number of vertices in a maximal irredundant set of G, respectively. Maximal Independent Set: An independent set S of vertices in a graph G is called a maximal independent set if S is not a proper subset of any other independent set of G. Maximal Irredundant Set: An irredundant set S of G is maximal irredundant if and only if S {v} is not irredundant for every v V G S. Maximum Degree: The maximum of the degrees of all the vertices in a graph G. Minimal Dominating Set: A dominating set S of a graph G is called a minimal dominating set if no proper subset of S is a dominating set of G. Minimum Degree: The minimum of the degrees of all the vertices in a graph G. Multipartite: An n partite graph is called multipartite if n > 2. n partite: A graph G is called n partite, n 2, if the vertex set may be partitioned into n subsets, such that no edge of G connects vertices from the same subset. Neighbours: If the unordered pair {u,v} = uv is an edge of the graph G, it is said that the vertices u and v are neighbours in G. Open Neighbourhood: The open neighbourhood of a vertex v in a graph G is the set of all vertices adjacent to v in G, and is denoted N(v). The open neighbourhood of a set S is defined as N[S] = {N[v] : v S}. Order: The cardinality of the vertex set of a graph G is called the order of G. viii

11 Product of two complete graphs: The product G = K m K n of two complete graphs K m and K n have vertex set V G = {v ij i = 1, 2,,m and j = 1, 2,,n} and edge-set E G = {{v ij,v kl } i = k and j l, or j = l and i k}. Private Neighbourhood: If S V G and s S, then the private neighbourhood of s relative to S, denoted by pn G (s,s), is the set N G [s] N G [S {s}]. Private Neighbours: The vertices of the private neighbourhood of S (pn G (s,s)) are called the private neighbours of s relative to S. Redundant: For S V G and s S, s is a redundant vertex of S if the private neighbourhood of s relative to S is empty. Symmetric: A graph that is vertex-transitive and edge-transitive. Semi-symmetric: A graph that is edge-transitive but not vertex-transitive. Singular Isolated Vertex: A vertex v S is a singular isolated vertex of S V G if pn G (v,s) = {v}. Size: The cardinality of the edge set of a graph G is called the size of G. Subgraph: A graph H is called a subgraph of G if V H V G and E H E G. Union: The union of two graphs H 1 and H 2, written as H 1 H 2, is the graph H with vertex set V H = V H1 V H2 and edge set E H = E H1 E H2. Universal Vertex: A vertex of G such that it is adjacent to all other vertices of G, is defined as an universal vertex. ix

12 Upper Domination Number: The maximum cardinality over all minimal dominating sets of a graph G is called the upper domination number of G, denoted Γ(G). Vertex: A vertex is a combinatorial element in terms of which a graph is defined. Vertices are indicated by nodes in the graphical representation of a graph. Vertex Set: The set comprised of all vertices of a graph G, is called the vertex set of G. Vertex-transitive: A graph G is vertex-transitive if for any u,v V G there exists an automorphism Φ of G such that Φ(u) = v. x

13 Table of Contents 1 Graph theoretic concepts Basic definitions Domination parameters Some basic results on domination parameters Criticality Vertex-transitivity and edge-transitivity Outline Vertex-criticality of the lower domination parameters Basic results on vertex-criticality The vertex-criticality of some classes of graphs Irredundance and Vertex-Criticality Edge-criticality of the lower domination parameters Basic results on edge-criticality The edge-criticality of some classes of graphs Irredundance and Edge-Criticality xi

14 4 ER-criticality of the lower domination parameters Basic results on π-er-criticality The π-er-criticality of some classes of graphs Results on ir-er-criticality π -ER-critical graphs Open Problems 59 References 61 Index 64 xii

15 Chapter 1 Graph theoretic concepts This chapter introduces the graph theoretic definitions required in this thesis, including some basic results on independence, domination and irredundance in graphs and their related parameters. For each of these parameters we define six types of criticality and discuss the existence and characterization of these types of criticality. 1.1 Basic definitions A graph G = (V G,E G ) is a finite, nonempty set of vertices V G, together with a (possibly empty) set of two-element subsets of V G, the edges E G, which is denoted by {u,v} = uv. The number of vertices in a graph G is called the order of G, while the number of edges in G is called the size of G. The open neighbourhood of v V G, denoted by N G (v), is the set {u V G uv E G } and the closed neighbourhood N G [v] is the set N G (v) {v}. The degree of v V G is the cardinality of the open neighbourhood of v and is denoted by deg G (v). The minimum degree δ(g) and the maximum degree (G) is, respectively, the minimum and maximum of the degrees taken over all the vertices of G. A vertex of degree 0 is an isolated vertex (it is adjacent to no other vertices of G) and a vertex of degree 1

16 V G 1 is a universal vertex of G (it is adjacent to all other vertices of G). A graphical representation of the graph G 1 with order 8 and size 6 is shown in Figure 1.1. From the figure we see that V G1 = {v 1,v 2,,v 8 } and E G1 = {v 1 v 2,v 2 v 5,v 5 v 7,v 7 v 6,v 6 v 3,v 3 v 4 }; while δ(g 1 ) = 0 and (G 1 ) = 2. Also, v 8 is an isolated vertex of G 1 and there exists no universal vertices in G 1. v 1 v 8 v 2 v 7 v 3 v 6 v 4 v 5 Figure 1.1: Graphical representation of the graph G 1 If uv E G, it is said that the vertices u and v are adjacent in G or that they are neighbours in G. With reference to the graph G 1 in Figure 1.1, we have v 1 adjacent to v 2, while v 4 and v 5 are not adjacent. Also, the neighbours of v 6 include v 7 and v 3. Two graphs G and H are isomorphic, denoted G = H, if there exists a bijection φ : V G V H such that uv E G if and only if φ(u)φ(v) E H. It is clear that if G = H, then from the definition of isomorphisms G = H. A graph that is isomorphic to a subgraph of G is also called a subgraph of G. The subgraph induced by a vertex-set S of G, denoted by G S, has vertex-set S and edge-set {uv E G u,v S}. A copy of G is a graph isomorphic to G. Two graphs G and H are disjoint if V G and V H are disjoint. The union G H of two graphs has V G H = V G V H and E G H = E G E H. Thus the disjoint union of G and H is the union of the disjoint copies of G and H, and the disjoint union of n copies of G will be denoted by ng. A graph G is connected if for any partition {V 1,V 2 } of V G there exists a v 1 V 1 and a v 2 V 2 such that v 1 v 2 E G ; otherwise G is disconnected. A component of G is a maximally connected subgraph of G. Suppose G is disconnected and let {V 1,V 2 } be 2

17 a partition of V G such that no vertices of V 1 are adjacent to any vertices of V 2. Then G is the disjoint union of the induced subgraphs G V 1 and G V 2 of G. Each of these subgraphs, if disconnected, can in turn be written as the disjoin union of two induced subgraphs. This procedure terminates in the decomposition of G into its components. 1.2 Domination parameters The closed neighbourhood N G [S] of a set S V G is the set s S N G[s], and the open neighbourhood N G (S) of a set S V G is the set N G [S] S. If S,T V G, then S dominates T in G if T N G [S] and if v N G [S], then we say S dominates v. If S V G and s S, then s is an isolated vertex of S in G if N G (s) S =, i.e. s is an isolated vertex of the graph G S. If S V G and s S, then the private neighbourhood of s relative to S, denoted by pn G (s,s), is the set N G [s] N G [S {s}]. The vertices of pn G (s,s) are called the private neighbours of s relative to S. If pn G (s,s) =, then s is a redundant vertex of S, otherwise it is an irredundant vertex of S. We refer to pn G (s,s) S as the external private neighbours of s relative to S and we denote it by epn G (s,s). If pn G (s,s) N[v] for s S and v V S, then we say that v annihilates s relative to S. S V G is an independent set of G if N G (s) S = for every s S, that is, no two vertices in S are adjacent in G. Observe that independence is a hereditary property, i.e. every subset of an independent set is independent. Consequently, an independent set S of G is maximal independent if and only if S {v} is not independent for all v V G S. The independence number β(g) is the largest number of vertices in a maximal independent set of G, and the lower independence number i(g) is the smallest number of vertices in a maximal independent set of G. 3

18 S V G is a dominating set of G if S dominates V G, i.e. every vertex of V G S is adjacent to at least one vertex of S. Domination is a super-hereditary property, as clearly every superset of a dominating set is dominating. It follows that a dominating set S of G is minimal dominating if and only if S {s} is not dominating for every S S. The domination number γ(g) and the upper domination number Γ(G) are the smallest and largest number of vertices in a minimal dominating set of G, respectively. S V G is an irredundant set of G if pn G (s,s) for every s S; thus every vertex of S is either isolated in S, or has an external private neighbour. As in the case of independence, irredundance is also a hereditary property and hence an irredundant set S of G is maximal irredundant if and only if S {v} is not irredundant for every v V G S. The irredundance number IR(G) and the lower irredundance number ir(g) are the largest and smallest number of vertices in a maximal irredundant set of G, respectively. In this thesis we only consider the lower domination parameters ir,γ,i. Also, all the theory related to a graph G assumes that G is a connected graph. For a disconnected graph we can just take the decomposition of components as described in Section 1.1, and apply the theory to each of the connected components, since π(g H) = π(g) + π(h) for π a lower domination parameter. Finally, by a π-set of G we mean a subset of V G realising π(g) for π {ir, γ, i}. 1.3 Some basic results on domination parameters In this section we briefly state some important results and bounds that will be used in Chapters 2, 3 and 4. The next proposition gives a characterization of maximal independence. Because of this characterization, the lower independence number i(g) of a graph G is also known as the independent domination number. 4

19 Proposition 1.1 (Berge [3]) S is a maximal independent set of G if and only if S is an independent dominating set of G. Cockayne and Hedetniemi [6] obtained the following characterization of minimal domination. Proposition 1.2 (Cockayne and Hedetniemi [6]) S is a minimal dominating set of G if and only if S is an irredundant dominating set of G. Using the concept of annihilation, Cockayne, Grobler, Hedetniemi and McRae [7] derived a characterization of maximal irredundance that states the following: Proposition 1.3 (Cockayne et al [7]) Suppose S is an irredundant set of G with U = V G N[S]. Then S is maximal irredundant if and only if for every v N[U] there exists an s v S such that v annihilates s v relative to S. The implication of this proposition is that for a given graph G and ir-set S of G, every v U annihilates some s S and every u N(U) must also annihilate some s S. We use this proposition in our study of graphs that are critical with respect to irredundance, since it helps us to define the structure of these graphs. Since we will only be examining the lower domination parameters of a graph G, we state the following well-known relationship. Proposition 1.4 (Cockayne, Hedetniemi and Miller [8]) For any graph G we have ir(g) γ(g) i(g) (1.1) 5

20 1.4 Criticality We are interested in how the lower domination parameters vary when the structure of the graph is slightly changed. For each of the lower domination parameters we define six types of criticality. For π {ir,γ,i}, the graph G is π-critical if π(g v) < π(g) for all v V G. π + -critical if π(g v) > π(g) for all v V G. π-edge-critical if π(g + uv) < π(g) for all uv E G. π + -edge-critical if π(g + uv) > π(g) for all uv E G. π-er-critical if π(g uv) > π(g) for all uv E G. π -ER-critical if π(g uv) < π(g) for all uv E G. Now we need to determine the existence of graphs with these types of criticalities. Firstly, all edgeless graphs with more than one vertex are π-critical and π-edge-critical for π {ir,γ,i}, while all stars K 1,n with n 1 are π-er-critical for π {ir,γ,i}. In [15], the following was shown: no π + -critical or π + -edge-critical graphs for π {ir,γ,i} exists, and no γ -ER-critical graphs exists. It was also shown that there do exist graphs which are i -ER-critical, but the existence of ir -ER-critical graphs is still an open question. Finally, we turn our attention to two useful results. In [15] Grobler proved the following proposition. We use this in proving under which circumstances certain classes of graphs are critical. Proposition 1.5 (Grobler [15]) ir(g) = γ(g) if and only if there exists an ir-set S of G and an x V G such that S {x} is a dominating set of G. 6

21 The following inequality was obtained by Allan and Laskar in 1978 and still proves to be a very useful relationship. Proposition 1.6 (Allan and Laskar [1]) For any graph G, γ(g) 2ir(G) Vertex-transitivity and edge-transitivity In this section we focus on vertex- and edge-transitivity. This will be important in our comparison of the different types of criticalities. Let us first define the basic terminology needed for this section, and then we will define the relationship between criticality and transitivity. In this section we will also briefly refer to Group Theory terminology which the reader can find in [14]. An automorphism of a graph G is an isomorphism of G onto itself. A graph G is vertextransitive if for any u,v V G there exists an automorphism φ of G such that φ(u) = v. This implies that G is vertex-transitive if and only if the group of all automorphisms of G acting on V G produces only one orbit. Also, from the definition of vertex-transitivity it follows that these automorphisms retain adjacency and non-adjacency between vertices and the between the neighbours of the vertices; hence N(u) = N(v) = k for all u,v V G and a fixed k. Thus a graph G is vertex-transitive if and only if deg G (v) = k for all v V G and each u,v V G is contained in the same cycles. A graph G is edgetransitive if for any u 1 u 2,v 1 v 2 E G there exists an automorphism φ of G such that φ({u 1,u 2 }) = {v 1,v 2 }. Similarly, this implies that G is edge-transitive if and only if the group of all automorphisms of G acting on E G produces only one orbit. A graph that is vertex-transitive and edge-transitive is called symmetric (for example K 3 ), while a graph that is edge-transitive but not vertex-transitive is called a semisymmetric graph. To obtain a semi-symmetric graph, we take any symmetric graph, except a cycle, and replace each edge with a path of two edges through a new vertex of degree 2. 7

22 An example of a class of graphs that is vertex-transitive but not edge-transitive is the product of K 2 with any symmetric graph, except K 1 and K 2. The next proposition shows an important relation between semi-symmetric graphs and bipartite graphs. As mentioned previously, we only consider connected graphs. Proposition 1.7 If a graph G is edge-transitive but not vertex-transitive, then it is bipartite. Proof: Suppose a graph G is edge-transitive but not vertex-transitive. Let the group Φ of all automorphisms of G act on V G. Since G is not vertex-transitive, this action produces more than one orbit. We now show that it produces exactly two orbits and that they form two independent sets that partition V G. Let A and B be two of the orbits. Consider any a A and b B. Since G is edgetransitive, it follows that for each x N(a) and y N(b) there exists a φ Φ such that φ({a,x}) = {b,y}. Therefore, for each x N(a) and y N(b) we have φ(a) = y and φ(x) = b; hence N(b) A and N(a) B. This holds for any a A and b B; hence A and B are independent sets of G and A B = V G. Thus it follows that a connected non-bipartite edge-transitive graph must be vertextransitive. The complement G of the graph G has V G = V G and E G = {{u,v} {u,v} / E G }. We now show that the complement of a vertex-transitive graph is also vertex-transitive. Proposition 1.8 If G is vertex-transitive, then G is vertex-transitive. Proof: Take any u,v V G. Then u,v V G. Since G is vertex-transitive, there exists an automorphism φ of G such that φ(u) = v. The automorphism group acting on a graph G is the same as the automorphism group acting on G, thus φ is also an automorphism of G such that φ(u) = v. Thus G is vertex-transitive. 8

23 Unlike vertex-transitivity, if G is edge-transitive, then G is not necessarily edge-transitive. The graph G = C 6 is edge-transitive (and vertex-transitive), but the complement G = K 2 C 3 is not, since there exists no automorphism φ such that φ({u,v}) = {w,x} for uv C 3 and wx K 2. Also, the graph H = K 2,5 is edge-transitive (but not vertextransitive), while the complement H is not edge-transitive. 1.6 Outline Since 1979, graphs that are critical with respect to domination have been thoroughly studied by Brigham, Chinn and Dutton [5], Sumner and Blitch [19] and Walikar and Acharya [20]. This was extended by Ao [2] to the study of graphs critical with respect to independence, and the study of irredundance critical graphs was initiated by Grobler [15]. Some open questions still remained, especially involving the characterization of graphs critical with respect to irredundance. The purpose of this thesis is to answer some of these questions. Chapter 2 deals with vertex-critical graphs. In Section 2.1 we present basic results concerning the lower domination parameters of vertex-critical graphs and the characterization of vertex-critical graphs in terms of singular isolated vertices. We also characterize vertex-critical graphs that are vertex-transitive. This is then implemented in Section 2.2 to determine the vertex-transitivity and vertex-criticality of four classes of graphs. This leads us to the formulation of some conjectures, which we examine further in Section 2.3. In Section 2.3 we also focus on determining necessary conditions for ir-critical graphs. In Chapter 3 we examine edge-critical graphs. In Section 3.1 we state results and relationships concerning the lower domination parameters of edge-critical graphs and the characterization of edge-critical graphs in terms of singular isolated vertices. We also characterize edge-critical graphs G such that G is edge-transitive. This is then implemented in Section 3.2 to determine the edge-criticality of three classes of graphs, leading 9

24 to some conjectures. In Section 3.3 we focus on the structure of ir-edge-critical graphs, and determine the validity of some of the conjectures in Section 3.2. Chapter 4 deals with edge-removal-critical graphs. In Section 4.1 we present the results and relationships of the π-er-critical graphs for π {i, ir, γ}, and characterize π-ercritical graphs for π {i,γ}. In Section 4.2 we determine which of the classes of graphs of Section 2.2 are π-er-critical for π {i,ir,γ}. In Section 4.3 we state the necessary conditions for a connected graph to be ir-er-critical and show that these graphs are neither vertex-transitive nor edge-transitive. We then discuss all the results thus far achieved in the literature with respect to the characterization of ir-er-critical graphs. In Section 4.4 we examine π -ER-critical graphs and list some results concerning the lower domination parameters of these graphs. We then finally use this knowledge to determine whether some classes of graphs are π -ER-critical for π {i,ir}. In Chapter 5 we conclude by briefly listing some remaining open questions and future recommended work. In all the chapters the results and proofs which are given without references are the original work of the author. 10

25 Chapter 2 Vertex-criticality of the lower domination parameters In this chapter we examine π-critical graphs for π {ir,γ,i}. From Chapter 1 we recall that a graph G is π-critical if and only if π(g v) < π(g) for all v V G, while G is π + -critical if and only if π(g v) > π(g) for all v V G. Also, the edgeless graphs K n with n 2 are π-critical, but it was shown by Grobler in [15] that there exist no π + -critical graphs for π {ir,γ,i}. 2.1 Basic results on vertex-criticality We define a k-π-critical graph to be a π-critical graph G such that π(g) = k for π a lower domination parameter. As mentioned in Section 1.6, graphs that are γ-critical were initially studied by Brigham, Chinn and Dutton in [5]. Some of their basic results included: Lemma 2.1 (Brigham, Chinn and Dutton [5]) The only 2-γ-critical graphs are nk 2 with n 1. 11

26 Lemma 2.2 (Brigham, Chinn and Dutton [5]) For any graph G and any v V G, γ(g v) γ(g) 1. This result implies that if G is a γ-critical graph, then γ(g v) = γ(g) 1 for all v V G. These three authors also presented some properties for n-γ-critical graphs, concerning bounds on the order of G in terms of (G), γ(g) and the size of G. Clearly G is γ- critical if and only if every component of G is γ-critical. Parallel to this, in [5] it was shown that G is γ-critical if and only if every block of G is γ-critical. Graphs that are i-critical were then studied by Suqin Ao [2] in her masters thesis, where she obtained results for i-critical graphs analogous to those in [5] for γ-critical graphs. Lemma 2.3 (Ao [2]) The only 2-i-critical graphs are nk 2 with n 1. Lemma 2.4 (Ao [2]) For any graph G and any v V G, i(g v) i(g) 1. She also obtained bounds on the order of n-i-critical graphs similar to those for n-γ-critical graphs, and showed that G is i-critical if and only if every block of G is i-critical. In her study of γ-critical and i-critical graphs, Lemmas 2.5 and 2.6 played a very important role. Lemma 2.5 (Ao [2]) If there exists vertices u,v V G such that N[v] N[u], then G is not γ-critical. Lemma 2.6 (Ao [2]) If there exists vertices u,v V G such that N[v] N[u], then G is not i-critical. Graphs that are ir-critical were first studied by Grobler in [15], where he obtained a result similar to those of Lemmas 2.1 and 2.3. Recall that π a lower domination parameter means that π {ir,γ,i}. 12

27 Proposition 2.1 (Grobler [15]) For π a lower domination parameter, the only 2-πcritical graphs are nk 2, n 1. We will see in Section 2.3 that a result similar to Lemmas 2.2 and 2.4 does not hold for ir-critical graphs. The next proposition gives a characterization of γ-critical and i- critical graphs in terms of singular isolated vertices. We define v V G as a singular isolated vertex of S V G if pn G (v,s) = {v}; thus v is an isolated vertex of S that has no external private neighbours relative to S. But first a lemma. Lemma 2.7 (Grobler [15]) Let π {γ,i}. For any graph G with more than one vertex, π(g v) = π(g) 1 if and only if v is a singular isolated vertex of some π-set of G. Proposition 2.2 (Grobler [15]) Let π {γ,i}. For any graph G with more than one vertex, (a) G is π-critical if and only if π(g v) = π(g) 1 for all v V G. (b) G is π-critical if and only if every vertex of G is a singular isolated vertex of some π-set of G. In the case of vertex-transitive graphs, we have Proposition 2.3 Let π {γ,i}. For any vertex-transitive graph G with more than one vertex, G is π-critical if and only if G has a π-set containing a singular isolated vertex. Proof: Assume G has a π-set T containing a singular isolated vertex v. Since G is vertex-transitive, there exists for any u V G an automorphism φ such that φ(v) = u. Thus φ(t) is a π-set of G with u a singular isolated vertex. As this is true for any u V G, it follows from Proposition 2.2(b) that G is π-critical. Finally, the following result shows an important relationship between i-critical and γ- critical and between γ-critical and ir-critical graphs. 13

28 Proposition 2.4 For any graph G, (a) if G is i-critical and i(g) = γ(g), then G is γ-critical. (b) if G is γ-critical and γ(g) = ir(g), then G is ir-critical. Proof: (a) Suppose G is i-critical and i(g) = γ(g). From Proposition 1.4 it follows that γ(g v) i(g v) < i(g) = γ(g) for all v V G. (b) Suppose G is γ-critical and γ(g) = ir(g). From Proposition 1.4 it follows that ir(g v) γ(g v) < γ(g) = ir(g) for all v V G. This proposition is integral in our proofs of vertex-criticality for the different classes of graphs with respect to their lower domination parameters. 2.2 The vertex-criticality of some classes of graphs In this section we consider four classes of graphs, namely the complete multipartite graphs, the product of two complete graphs, the complement of the product of two complete graphs and the circulants. In [15] Grobler determined the lower domination parameters for each of these classes. By applying the theory given in Chapter 1 and Section 2.1 and these values for the lower domination parameters, we now determine which of these classes of graphs are vertex-critical. The complete multipartite graph K n1,n 2,,n m is the complement of the disjoint union K n1 K n2 K nm. Thus the vertex-set of K n1,n 2,,n m has partition {V 1,V 2,,V m } with V i = n i for 1 i m, and uv is an edge of K n1,n 2,,n m if and only if u and v do not belong to the same partite set. The graph K 4,4 is shown in Figure

29 Clearly, K n1 K n2 K nm for m 2 is vertex-transitive if and only if n 1 = n 2 = = n m. Thus from Proposition 1.8 it follows that K n1,n 2,,n m for m 2 is vertex-transitive if and only if n 1 = n 2 = = n m. v 1 v 5 v 2 V 1 V 2 v 6 v 3 v 7 v 4 v 8 Figure 2.1: K 4,4 Proposition 2.5 (Grobler [15]) If G = K n1,n 2,,n m with m 2, then 2 if n i > 1 for all i = 1, 2,,m ir(g) = γ(g) = 1 otherwise i(g) = min{n i : i = 1, 2,,m} Now we determine which of the complete multipartite graphs are vertex-critical, using the values of the lower domination parameters as given above. Proposition 2.6 Let G = K n1,n 2,,n m with m For π {ir,γ}, G is π-critical if and only if n 1 = n 2 = = n m = G is i-critical if and only if n 1 = n 2 = = n m 2. 15

30 Proof: If n i = 1 for some i = 1, 2,,m, then π(g) = 1; hence G is not π-critical for π {i,ir,γ}. Assume therefore that n i > 1 for all i = 1, 2,,m. 1. By Proposition 2.5, if π {ir,γ}, then π(g) = 2. Hence by Proposition 2.1, G is π-critical if and only if G = K 2,2,,2. 2. If n 1 = n 2 = = n m = n (say), then i(g) = n and i(g v) = n 1 for all v V G by Proposition 2.5. If n k > n j for some k j, then i(g) = min{n i : i = 1, 2,,m} = i(g v) for v V k. From this proposition it follows that K n1,n 2,,n m is an example of a graph that is i-critical, but neither γ-critical nor ir-critical for n 1 = n 2 = = n m 3. Next we turn our attention to the class of graphs defined as the product of two complete graphs. The product K m K n of two complete graphs K m and K n have vertex set V = {v ij i = 1, 2,,m and j = 1, 2,,n} and edge-set E = {{v ij,v kl } i = k and j l, or j = l and i k} Let X i = {v ik k = 1, 2,,n} and Y j = {v kj k = 1, 2,,m} for each i = 1, 2,,m and j = 1, 2,,n. Refer to Figure 2.2 for an illustration of K m K n for m = n = 3. It is easy to see that G = K m K n with n m 2 is vertex-transitive, since there exist an automorphism φ 1 of G such that φ 1 (v i,j ) = v i,j+1 and φ 1 (v i,n ) = v i,1 for each 16

31 i = 1, 2,,m and an automorphism φ 2 of G such that φ 2 (v i,j ) = v i+1,j and φ 2 (v m,j ) = v 1,j for each j = 1, 2,,n. Y 1 Y 2 Y 3 X 1 v 11 v 12 v 13 X 2 v 21 v 22 v 23 X 3 v 31 v 32 v 33 Figure 2.2: K 3 K 3 Proposition 2.7 (Grobler [15]) If G = K m K n with n m 2, then ir(g) = γ(g) = i(g) = m The following result shows that the product of two complete graphs is vertex-critical for all the lower domination parameters under the assumption that n = m. Proposition 2.8 Let G = K m K n with n m 2. Then G is π-critical, for π {i,ir,γ}, if and only if n = m. Proof: Suppose m = n. Then the i-set consisting of the diagonal vertices {v ii : 1 i n} contains a singular isolated vertex, namely v 11. Since G is vertex-transitive it follows from Proposition 2.3 that G is i-critical, and from Propositions 2.4((a) and (b)) and 2.7 it then follows that G is also ir-critical and γ-critical. Suppose now that m < n. We show that there is no γ-set S that contains a singular isolated vertex. Suppose without loss of generality that v 11 is a singular isolated vertex of 17

32 some γ-set S of G. Then Y j S for j = 2,,n. Therefore S n. This contradicts S = m < n. It follows that no γ-set of G exists such that v 11 is a singular isolated vertex of that set. Thus by Proposition 2.2, G is not γ-critical, and from Proposition 2.4 it follows that G is also not i-critical. Thus γ(g v) = i(g v) = m for any v V G. We now show that ir(g v) = m for any v V G. Since G is vertex-transitive, we can assume without loss of generality that ir(g v mn ) = p < m. Let H = G v mn and consider an ir-set S of H. Since γ(h) = m, S is not a dominating set of H. Let u = v kl be a vertex of H not dominated by S. Then, since S is maximal irredundant, u annihilates some non-isolated vertex s = v ij of S. Therefore pn H (s,s) = {v il } or {v kj }. Now if m = 2, then p = 1 and the graph is not ir-critical. Thus n > m 3 for i k, j l. If pn H (s,s) = {v il }, then S Y t for 1 t n 1, t l and S Y j 2. Therefore S n 1 m, contradicting ir(h) < m; hence pn H (s,s) = {v kj }. If j < n, then S X t for 1 t m, t k and S X i 2. Therefore S m, contradicting ir(h) < m. Thus j = n, i.e. s Y n. Now S X t for 1 t m 1, t k and S X i 2; hence S m 1 and therefore S = m 1. It follows that S X m = and therefore w = v ml is also not dominated by S. Therefore w annihilates some non-isolated vertex t of H. By using the same argument as above, t Y n, which is impossible since pn H (s,s) = {v kn }. We next consider the complement of the product of two complete graphs (for example Figure 2.3). From Proposition 1.8 it follows that K m K n is vertex-transitive for any n m 2. In [15] the following values for the lower domination parameters of these graphs were obtained: Proposition 2.9 (Grobler [15]) If G = K m K n with n m 2, then ir(g) = γ(g) = min{3,m} and i(g) = m 18

33 X 1 Y 1 Y 2 Y 3 v 11 v 12 v 13 X 2 v 21 v 22 v 23 X 3 v 31 v 32 v 33 Figure 2.3: K 3 K 3 This enables us in the following proposition to determine under which conditions the complement of the product of two complete graphs is vertex-critical. Proposition 2.10 Let G = K m K n with n m For π {ir,γ}, G is π-critical if and only if n m = G is i-critical if and only if n m 3. Proof: Let m = 2. If π {i,ir,γ}, then from Proposition 2.9 we have π(g) = 2. Thus it follows from Proposition 2.1 that G is not π-critical. Now assume m 3. The i-set {v 11,v 21,,v m1 } contains m singular isolated vertices. Since G is vertex-transitive, it follows from Proposition 2.2 that G is i-critical. Furthermore, if m = 3 it follows from Proposition 2.4 that G is also ir-critical and γ-critical. Suppose now m 4. We want to show that G is neither ir-critical nor γ-critical. From Proposition 2.9 we know that ir(g) = γ(g) = 3. Without loss of generality remove v mn. Since H = G v mn contains no universal vertices, ir(h) > 1; hence assume ir(h) = 2 and let S = {s 1,s 2 } be an ir-set of H. From Proposition 2.9 we have i(g) = m, and since G is i-critical, it follows that i(h) = m 1 > 2 = ir(h); hence S is not independent; hence 19

34 s 1 = v ij and s 2 = v kl, with i k and j l. Then S {v xy } with v xy N(s 1 ) N(s 2 ) is still an irredundant set in H, since pn H (s 1,S {v xy }), pn H (s 2,S {v xy }) and pn H (v xy,s {v xy }). Thus S is not maximal irredundant as previously assumed; thus G is not ir-critical, and since ir(g) = γ(g), it follows from Proposition 2.4 that G is not γ-critical. It follows that if n m 4, then G = K m K n is another example of a class of graphs that is i-critical but neither γ-critical nor ir-critical. Lastly, we define the circulant C n a 1,a 2,,a l with 0 < a 1 < a 2 < < a l < n by specifying the vertex and edge sets V = {v 1,v 2,,v n } and E = {{v i,v i+j } : i = 1, 2,,n and j = a 1,a 2,,a l } Consider now the circulant C n 1, 2,,r for n 3, 1 r n 2. For each vi V let N[v i ] = {v i r,,v i 1,v i,v i+1,,v i+r }. In Figure 2.4 a graphical representation of C 11 1, 2 is given. v 1 v 11 v 2 v 10 v 3 v 9 v 4 v 8 v 5 v 7 v 6 Figure 2.4: C 11 1,2 20

35 Now we determine under which conditions this specific class of circulants is vertex-critical. But first the lower domination parameters of these graphs. Proposition 2.11 (Grobler [15]) If G = C n 1, 2,,r with n 3, 1 r n 2, then n ir(g) = γ(g) = i(g) = 2r + 1 From this proposition we clearly see that if r = n 2, then G = Kn. Then ir(g) = γ(g) = i(g) = 1; hence G is not π-critical for π {i,ir,γ}. Therefore we need only examine the vertex-criticality of the circulant G = C n 1, 2,,r for 1 r < n 2. Also, G = C n 1, 2,,r with n 3 and 1 r < n 2 is vertex-transitive, since degg (v) = 2r for all v V G and each vertex of G is contained in the same cycles. Proposition 2.12 Let G = C n 1, 2,,r with n 3 and 1 r < n 2. Then G is π-critical for π {i, ir, γ} if and only if n 1 mod(2r + 1). Proof: By the division algorithm, there exists unique integers m and q with 0 q 2r such that n = (2r + 1)m + q. First we assume q = 1 and show that G is π-critical for π {i,ir,γ}; then we assume G is π-critical and show that this holds only if q = 1. Assume q = 1. Then n = (2r + 1)m + 1; hence ir(g) = i(g) = γ(g) = n 2r+1 = m + 1. If m = 1, then n = 2r + 2 and G = C 2r+2 1, 2,,r = (r + 1)K 2. Therefore from Proposition 2.1 it follows that G is 2-π-critical for π {i,ir,γ}. Let m 2. The set S = {v 1,v 1+(2r+1),v 1+2(2r+1),,v 1+(m 1)(2r+1) } of G is independent and dominates all vertices of G except u = v n r. Therefore S {u} 21

36 is an independent dominating set of G with u a singular isolated vertex. Since G is vertex-transitive, it follows from Proposition 2.2 and Proposition 2.4 that G is π-critical. Let us now assume G is π-critical, and show that q = 1. We want to show that q 0, so assume to the contrary that q = 0. Then i(g) = γ(g) = ir(g) = m from Proposition 2.11 and n = (2r + 1)m. Therefore S = {v 1,v 1+(2r+1),v 1+2(2r+1),,v 1+(m 1)(2r+1) } is an i-set of G. Since G is i-critical, m 2 and i(g v 2r+1 ) = m 1. Let S be an i-set of G v 2r+1. Then the n 1 = (2r + 1)m 1 vertices of G v 2r+1 are not dominated by the m 1 vertices of S, since m 1 vertices dominates at most (m 1)(2r + 1) = (2r + 1)m 1 2r < (2r + 1)m 1 vertices of G v 2r+1. Hence S is not a dominating set of G v 2r+1, which contradicts q = 0. Thus q 0; hence from Proposition 2.11 it follows that i(g) = γ(g) = ir(g) = m + 1. If m = 1, then n = 2r q and since G is 2-π-critical it follows from Proposition 2.1 that G = C n 1, 2,,r = ik 2 for i 2. Since i(g v) = 1 for all v V G, it follows that deg v (G) = n 1 = 2r + 1 for all v V G. Hence q = 1. Now let m 2. Assume v 1 V G is a singular isolated vertex of some i-set S of G. Then {v n r,v 1,v 2+r } S is such that {v n r,v 1,v 2+r } dominates 2(2r + 1) + 1 vertices of G. Therefore the remaining (m 2)(2r + 1) + q 1 vertices of G must be dominated by the remaining m 2 vertices of S. Hence (m 2)(2r + 1) + q 1 (m 2)(2r + 1) = q 1 vertices are not dominated by S, but since S is an dominating set of G, it follows that q = 1. From this proposition it follows that C 11 1, 2 (see Figure 2.4) is an example of a π-critical circulant for π a lower domination parameter. It is interesting to note that in all four classes of graphs, irredundance meets the require- 22

37 ments of Proposition 2.2, namely that a graph G is ir-critical if and only if every vertex of G is a singular isolated vertex of some ir-set of G, and that in all these examples ir(g) = γ(g). This motivates the following two conjectures. Conjecture 2.1 For any ir-critical graph G, ir(g) = γ(g). Conjecture 2.2 For any graph G, G is ir-critical if and only if every vertex of G is a singular isolated vertex of some ir-set of G. The open question still remains whether Proposition 2.2 will still be true for π = ir if ir(g) γ(g). In Section 2.3 we manage to show that given an ir-critical graph G, ir(g) = γ(g) for k Irredundance and Vertex-Criticality We start this section with a useful result by Favaron. We include a shortened proof. Remember that all graphs referred to are connected; for disconnected graphs, the propositions can just be applied to their components. Theorem 2.1 (Favaron [12]) For any graph G with v V G and ir(g v) 2, we have ir(g) 2ir(G v) 1 Proof: First note that ir(g v) + 1 2ir(G v) 1 if and only if ir(g v) 2. Let A be an ir-set of the graph G v. Then A is irredundant in G. Thus there exists a maximal irredundant set A of G such that A A. If A A 1, then ir(g) A = A + A A ir(g v) + 1 2ir(G v) 1 23

38 So assume A A 2 and let y A A with y v. Then A {y} is irredundant in G, but not in G v. Thus there exists an x A {y} such that pn G v (x,a {y}) = but pn G (x,a {y}) ; hence pn G (x,a {y}) = {v}. This implies that A {y,v} is not an irredundant set in G, thus v A. Since A A 2, it then follows that x A and v pn G (x,a). Let U be the set of vertices in G v not dominated by A and let B be the set of non-isolated vertices of A in G v which are annihilated by vertices in U. If B {x}, let D consist of all vertices in A except for some t B {x}, together with one private neighbour of each non-isolated vertex of A in the graph G v. Since each vertex of N G v [U] annihilates some non-isolated vertex of A, D dominates N G v [U]. Furthermore, t is non-isolated and thus dominated by A {t}, and v is dominated by x D. Therefore D is a dominating set of G with D 2 A 1; hence ir(g) γ(g) 2ir(G v) 1 If B = {x}, let D consist of all vertices of A together with one private neighbour of x in the graph G v. Since each vertex of N G v [U] annihilates x, and x D dominates v, it follows that D is a dominating set of G with D = A A 1; hence ir(g) γ(g) 2ir(G v) 1 It is possible to construct a graph G such that ir(g) = 2ir(G v) 1 holds. We examine any graph G and remove a vertex v. Let S = {v 1,v 2,v 3,,v k } be an ir-set of the graph G v, {u i } be the external private neighbourhood of v i for i = 1, 2,,k, and let N(v i ) N(v j ) = m 2k for i,j = 1, 2,,k, j i. Also each u i for i = 1, 2,,k has m private neighbours (refer to Figure 2.5 for k = 3 and m = 6). Then by taking the ir-set (S v k ) {u 1,u 2,,u k }, we have ir(g) = 2k 1. Thus we have shown that a 24

39 result similar to Lemmas 2.2 and 2.4 does not hold for irredundance. v 1 u 1 v v 2 u 2 v 3 u 3 Figure 2.5: A graph G such that ir(g) = 2ir(G v) 1 Now we show that Conjecture 2.2 is true for 2-ir-critical graphs. Proposition 2.13 G is 2-ir-critical if and only if every vertex of G is a singular isolated vertex of some ir-set of G. Proof: For π {i,ir,γ}, the only 2-π-critical graphs are nk 2 (Proposition 2.1), and every vertex of nk 2 is clearly a singular isolated vertex of an ir-set of G. From Proposition 1.6 we already know that, if ir(g) = k, then γ(g) 2k 1. We next show that, if G is k-ir-critical, then this bound can be improved to γ(g) 2k 2. Even though Proposition 2.14 is true for k = 2, we characterized 2-ir-critical graphs in Proposition 2.13, and now only concentrate on k 3. Proposition 2.14 If G is k-ir-critical with k 3, then γ(g) 2k 2. Furthermore, if γ(g) = 2k 2, then G is also (2k 2)-γ-critical with γ(g v) = 2k 3 and ir(g v) = k 1 for all v V G. Proof: By Lemma 2.7 and Proposition 1.6, and the k-ir-criticality of G, γ(g) 1 γ(g v) 2ir(G v) 1 2k 3 (2.1) 25

40 for all v V G. Therefore γ(g) 2k 2 and, if γ(g) = 2k 2, then equation (2.1) reduces to 2k 3 γ(g v) 2ir(G v) 1 2k 3; hence γ(g v) = 2k 3 and ir(g v) = k 1 for all v V G. From Proposition 2.2 it then follows that G is (2k 2) γ-critical. The next proposition gives necessary conditions for a graph G to be k-ir-critical with k 3 and γ(g) = 2k 2. We will use the following notations in its proof: Take any v V G and suppose S = {v 1,v 2,,v k 1 } is an ir-set of G v. Let U, P i and C denote the (possibly empty) sets of vertices in V G v S which are adjacent to no vertices, exactly one vertex v i, and at least two vertices of S. Thus U = V G v N G v [S] P i = epn G v (v i,s) for i = 1, 2,,k 1 k 1 C = N G v (S) ( P i ) Furthermore, let A i for i = 1, 2,,k 1 denote the (possibly empty) set of vertices in U that annihilates only the vertex v i and A denote the (possibly empty) set of vertices in U that annihilates two or more vertices of S. We can see that the sets S, U, P i and C form a disjoint partition of V G v (since every vertex in V G v is either in S, or adjacent to no vertices of S, or adjacent to exactly one vertex of S, or adjacent to two or more vertices of S). If we then denote the isolated vertices of S in G v by I, the non-isolated vertices of S in G v that are annihilated by some u U by B, and B = S (I B), it also follows that S is partioned into the disjoint sets I, B and B with S = I + B + B = k 1. i=1 26

41 Proposition 2.15 Given a k-ir-critical graph G with γ(g) = 2k 2, k 3. Then, for any v V G and an ir-set S of G v, 1. S = B. 2. P i = epn G v (v i,s) for all i = 1, 2,,k v is not adjacent to any vertex of S P 1 P 2 P k A i for all i = 1, 2,,k v does not dominate A i for any i = 1, 2,,k C N G v [U]. 7. {v,p 1,p 2,,p k 1 } does not dominate C N G v [U] for any p i P i, i = 1, 2,,j. Proof: Let G be a k-ir-critical graph with γ(g) = 2k 2, k 3. For any v V G, let S be an ir-set of G v. Take any v i S for i = 1, 2,,k 1 and let H i be the set of vertices in G v consisting of S {v i } together with one external private neighbour for each vertex in B. It follows from the definition of ir-criticality that every vertex in N[U] must annihilate some vertex of S, thus every p i P i is adjacent to some P j, j = 1, 2,,k 1. Thus by choosing one external private neighbour for each vertex in B, we ensure that every P i is dominated. Thus the set H i, with H i 2k 3, is a dominating set of G v, and since γ(g v) = 2k 3 from Proposition 2.14, it follows that H i = 2k 3; hence H i is a γ-set of G v. Also, H i = 2k 3 implies that S = B = k 1. Thus P i = epn G v (v i,s) for i = 1, 2,,k 1. Also, since γ(g) = 2k 2, it then follows that v is not adjacent to any vertex of S P 1 P 2 P k 1, otherwise the set H i would be a dominating set of G for some i = 1, 2,,k 1. Let H be the set of vertices in G v consisting of S {v 1 } together with one external private neighbour for each vertex in B v 2. Thus A 2, otherwise the set H with H = 2k 4 < γ(g v) is a dominating set of G v. Also, v does not dominate A 2, 27