RECONSTRUCTING PROPERTIES OF GRAPHS

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1 RECONSTRUCTING PROPERTIES OF GRAPHS S.Jerlin Mary Assistant Professor, Nanjil Catholic College Of Arts & Science Kaliyakkavilai,K.K.Dist, TamilNadu,India Abstract:This paper discuss the basic definitions and results in reconstructible graphs. Also discuss about card, edge card,dacard, decks, edge deck, reconstruction, edge reconstruction, class reconstruction, reconstruction number, edge reconstruction number, degree associated reconstruction number and class reconstruction number and the reconstruction conjecture of Kelly[1] and Ulam[2]. Also discuss the reconstruction number logic, regular graphs are reconstructible, disconnected graphs are reconstructible, number of edges of G is reconstructible, degree sequence of G is reconstructible and some important theorems in reconstructible graphs. I. INTRODUCTION The study on reconstruction number of graphs is an active area of research in graph theory. This paperdiscuss some reconstruction problems for graphs. A subgraph of a graph G obtained by deleting a vertex v and all edges incident with v is called a vertex deleted subgraph or a card of G. The deckd(g) of Gis the collection of all its cards. The graph H said to be a reconstruction of G if D(H)=D(G). A graph G said to be reconstructible if every reconstruction of G is isomorphic to G. A parameterθ=θ(g) is said to bereconstructible if for all reconstructions H of G,θ(H)=θ(G). Reconstruction Conjecture(Ullam[8]):Every finite graph with atleast three vertices is reconstructible. For a reconstructible graph, Harary and Plantholt[3] have defined the reconstruction number rn(g) to be minimum number of cards which can only belong to the deck of G and not to the deck of any other graph H, H not isomorphic to G; thus uniquely identifying the graph G. Let G be a graph and let s be a positive integer less than rn(g), ans-blocking set of G is a family Fof graphs such that G is not in Fand any s cards in the deck of G will also appear in the deck of some graph of F. II. RECONSTRUCTIBLE PARAMETERS Definition:A subgraph of a graph G obtained by deleting a vertex u and its incident edges from G is called vertex deleted subgraph or card of G. The collection of all the vertex deleted subgraphs {G-v: vϵv(g)} is called the deck of G and is denoted by D(G). The graph H is said to be a reconstruction of G if D(H)= D(G). A graph G is said to be reconstructible if every reconstruction of G is isomorphic to G. The following conjecture is proposed by S.M.Ulam in Reconstruction Conjecture(Ulam[8]): All graphs on at least three vertices are reconstructible. Remark: The graph G=K2 and H=2K1 are reconstructions of one another. But they are not reconstructible, since they are not isomorphic. Definition: A parameterθ = θ(g) is said to be reconstructible if for all reconstruction H of G, we have θ(g) = θ(h). In other words θ(g) is reconstructible if it can be determined uniquely from the deck of All Rights Reserved 34

2 Lemma: The number of vertices of a graph G is reconstructible. Proof: Take one card G-v of the graph G. Clearly G-v contains all vertices of G except v and so V(G) = V(G-v) +1. Lemma: The number of edges of a graph G is reconstructible. Proof: Consider the deck D(G) of a graph G of order n and size m. Then E(G-v) is the number of edges in the card G-v in D(G), vϵv(g). Thus E(G-v) is the sum of the edges of all cards in D(G). Let v1v2 be an arbitrary edge in G. Then this edge v1v2 does not lie in the two cards G-v1 and G-v2 and it lies in all other n-2 cards. Since the edgev1v2 is arbitrary, m= E(G v i) n 2. Since the RHS of the above equation is known from D(G), RHS is constructible. Thus the number of edges of a graph is reconstructible. Example: Consider the deck D(G) of some graph G given below. E(G-vi) =8. Each edge in G appears exactly n-2=4-2=2 cards in D(G). 1 Therefore the number of edges m of G is n 2 E(G-vi) =1(8)=4. 2 Theorem: The degree sequence of a graph G is reconstructible. Proof: Given a card G-v in the deck of G, the degree of v in G is clearly m- E(G-v). Since E(G-v) is known from G-v and m is reconstructible, it follows that the degree of v in G isreconstructible. Thus the degree sequence of G is reconstructible. Example: Consider the deck D(G) of some graph G shown below. The total number of edges m, in G is 4. Hence deg G (v 1 )=m- E(G-v 1 ) =4-1=3, deg G (v 2 )=m- E(G-v 2 ) =4-3=1, deg G (v 3 )=m- E(G-v 3 ) =4-2=2, deg G (v 4 )=m- E(G-v 4 ) =4-2=2. Thus the degree sequence of G is (3,1,2,2). Theorem: The neighbourhood degree sequence of a vertex v, NDS(v), in a graph G is reconstructible. Proof: Consider the card G-v in the deck D(G). To find the degrees of the neighbours of v in G; let d be the degree sequence of G-v but with the degree of v inserted in its correct position. The non-zero entries of the vector difference DS(G)-d occur in positions corresponding to the neighbours of v in G and their degrees can be read of from All Rights Reserved 35

3 Hence NDS(v) is reconstructible. Lemma: (Kelly s Lemma): Suppose F and G are graphs with V(F) < V(G). Then s(f,g) is reconstructible from the deck of G. Proof: Let G and F be two graphs with n and m vertices respectively such that m<n. Then s(f,g-v i ) counts all subgraphs isomorphic to F in the deck of G. Now let H be a subgraph of G isomorphic to F. Then V(H) =m. Clearly H does not lie in the cards of G obtained by deleting the m vertices of H from G. Hence H occurs in exactly n-m cards of G. Since H is arbitrary, each subgraph of G isomorphic to F occurs in exactly V(G) - V(F) cards in D(G). Hence s(f,g)= s(f,g v i). Since the RHS is known fromd(g), it follows that s(f,g) is reconstructible. V(G) V(F) Example:Consider the deck D(G) of some graph G given below. S(F,G-v 1 )=0, s(f,g-v 2 )=1, s(f,g-v 3 )=0, s(f,g-v 4 )=0. Thus s(f,g-v i )=1. As F =3, the subgraph F occurs in exactly one card in D(G). Hence s(f,g)=1. Theorem: Regular graphs are reconstructible. Proof: Let F be the family of regular graphs. Let D(G) be the given deck of some graph G. Since the degree sequence of G is reconstructible, if deg G (v)=k for all vϵv(g), then G is k-regular. Hence F is recognizable. Now consider a card G-v in the deck D(G). Since G is k-regular, G-v contains exactly k vertices of degree k-1 and the rest of the vertices have degree k. Now G can be obtained uniquely by adding a new vertex to G-v and joining it with all the k-vertices of degree k-1 in G-v. Therefore regular graphs are reconstructible. Example: Given D(G)={G-v i /i=1,2,3,4,5,6}. The graph G can be obtained from the card G-v 1 by adding a new vertex v and joining v to the vertices v 2 and v 3 of degree All Rights Reserved 36

4 Theorem: A graph G is reconstructibleiff the complement G is reconstructible. Proof: Assume that G is reconstructible. To prove that G is reconstructible. Given the deck ofg, D(G )={G -v i : i=1,2,,n} Then {G, v 1,G v n }={G-v 1,., G-v n }=D(G). ThereforeD(G) is known. Hence by our assumption G can be obtained uniquely from D(G). Hence G is known. That is G is reconstructible. The converse part follows from the fact that G =G. Theorem: Disconnected graphs are reconstructible. Proof: We proceedby two cases depending upon whether G has an isolated vertex or not. Case:1: G has isolated vertices. Now the minimum degree is zero. It can be determined from D(G), since the degree sequence is reconstructible. Also the card G-v obtained by deleting an isolated vertex from G can be identified in D(G). Hence G can be reconstructed by adding an isolated vertex w to this G-v. Case:2: G has no isolated vertices. We know that the class of disconnected graphs is recognizable. Therefore, we can assume that G is disconnected. Now, among all the components of all the cards in D(G), let C be a component of G with maximum number of vertices, Now label a non-cut vertex of C by v 0. Choose a card G-v in D(G) such that G-v contains maximum number of components isomorphic to C. Then the card C-v should have obtained from G by deleting a vertex similar to v 0 from a component isomorphic to C. Therefore, G can be obtained uniquely from G-v by replacing the component C-v 0 by C. Hence disconnected graphs are reconstructible. Definition: For a reconstructible graph G, the reconstruction number rn(g) is the minimum number of cards needed to identify the graph upto isomorphism. Definition: A subgraph of G obtained by deleting an edge e from G is called an edge-card of G. The collection of all the edge cards {G-e/eϵE(G)} is called an edge deck of G and is donoted by ED(G).The graph H is said to be an edge reconstruction of G if ED(G)=ED(H). A graph G is said to be edge reconstructible if every edge reconstruction of G is isomorphic to G. Edge Reconstruction Conjecture (Harary[3]): All graphs on atleast four edges are edge reconstructible. Definition: A familyc of graphs, the C reconstruction number or class reconstruction number Crn(G) of a graph GϵC is the minimum number of cards from its deck needed to identify G given the knowledge that GϵC; that is, G is the only graph in C having multiset of cards in its deck. Definition: A degree associated card or dacard of a graph (or digraph) is a pair (C,d) consisting of a card C in the deck and the degree (or in/out degree pair) d of the deleted vertex. The multiset of dacards is the dadeck (the degree associated deck). Definition: Degree associated reconstruction number drn(g) of a graph G to be the minimum number of dacards required to identify the graph G. Result:Myrvold[15] proved thatrn(g)=3 for every tree with more than two vertices other than P 4 while drn(p 4 )=2=rn(G)-2. Result:Myrvold[13] and Bollobas[2] proved that rn(g)=3 forr almost every All Rights Reserved 37

5 Result: Harary and Lauri[5] proved that the class reconstruction number of tree T is atmost 3(the resultof Myrvold[15] that rn(t) 3 strengthensthis) and they conjectured that the class reconstruction number of every tree is atmost 2. Result: For any graph G, drn(g) rn(g). Lemma: For any graph G, drn(g)=drn(g ). Proof: Let v be a vertex in an n-vertex graph G. Since d G (v)=n-1-d G (v) and G v =G -v, it follows that (C,d) is a dacard of G iff (C, n-1-d) is a dacard of G. Also G and G determine each other. Hence we conclude that the dacards of G from a vertex subset S determine G iff the dacards of G from S determine G. Theorem: If G is a k-regular graph on n-vertices then drn(g) min{k+2, n-k+1}. Proof: Let G be a k-regular graph on n-vertices. Then G is n-1-k regular. Sincedrn(G)=drn(G ) it satisfies to prove that drn(g) k+2. Let H be a graph that shares k+2 dacards with G. Let (C,k) be one shared dacard, so C=H-u for some uϵv(h). Since C also arises by deleting one vertex from the k-regular graph G, the graph C has k-vertices of degree k-1 and n-1-k vertices of degree k. Attaching u to the k-verties of degree k-1 in C forms a copy of G. If H is not isomorphic to G, then some vertex v in the neighbourhood of u in H has degree k in H whose deletion produces a card of G must be adjacent in H to every vertex of degree k+1 in H. There can be atmost k+1 such vertices, which contradicts the assumption of k+2 shared dacards with G. Hence G is isomorphic to H. III. RECONSTRUCTION NUMBER LOGIC 1)Get all cards of a given deck. 2)Entend the cards. 3)The intersection of the extended cards is the solution. IV. FUTURE WORKS The definition of rn(g), ern(g), drn(g), Crn(G) are very easy to understandable but it is very difficult to find the exact values so the future work is Is there any possibilities to find these values using rank of a matrix that represents the graph G or dimensions of the vector space which represents G etc. REFERENCES 1. Kelly P. J., A Congruence Theorem for Trees, Pacific J. Math, 7, , Ulam S. M., A Collection of Mathematical Problems, Wiley (Interscience), New York, F. Harary and M. Plantholt. The graph reconstruction number. J. Graph Theory, 9(4): , All Rights Reserved 38