CLASSES OF VERY STRONGLY PERFECT GRAPHS. Ganesh R. Gandal 1, R. Mary Jeya Jothi 2. 1 Department of Mathematics. Sathyabama University Chennai, INDIA

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1 Inter national Journal of Pure and Applied Mathematics Volume 113 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu Abstract: CLASSES OF VERY STRONGLY PERFECT GRAPHS Ganesh R. Gandal 1, R. Mary Jeya Jothi 2 1 Department of Mathematics Sathyabama University Chennai, INDIA 2 Department of Mathematics Sathyabama University Chennai, INDIA A graph G is Very Strongly Perfect Graph if for every induced Sub graph H every vertex of H belongs to an independent set of H meeting all maximal cliques of H. The Structure of very strongly perfect Graphs have been characterized by some classes of graphs like bipartite graph, complete bipartite graph, wheel graph, gear graph etc. AMS Classification - MS Key Words: very strongly perfect graphs, independent set, cubic bipartite graph. gear graph, wheel graph. 1. Introduction In this paper, graphs are finite and simple, that is they have no parallel edges and loops. Let G(V, E) be a graph. A clique C in a undirected graph G is a complete sub graph. A independent set is a set of vertices of graph G, no two of which are adjacent. The number of vertices in largest independent set of graph G(V, E) called independence number denoted by Ψ(G). A path in a graph G(V, E) is a alternating sequence of vertices and edges. A path with terminal vertices are coinciding called cycle and it is denoted by C n. The number of vertices in C n equals to the number of edges. The even cycle is a cycle with even number of vertices and the odd cycle is a cycle with odd number of vertices. A bipartite graph is a graph whose vertex set V can be partitioned into two disjoint sets V 1 and V 2 such that every edge e = uv has one end in V 1 and other in V 2. A complete bipartite graph G is an bipartite graph with partition V = V 1 V 2 such that for any two vertices v 1 V 1 and v 2 V 2, v 1 v 2 is an edge in G and V 1 V 2 = φ.the complete bipartite graph with partition of size V 1 = m and V 2 = n is denoted by K m,n. A tree is the connected acyclic graph. A wheel graph W m, m 3 is a graph obtained by joining all vertices of cycle C m to a further vertex c called centre. A gear graph G is obtained from wheel graph by adding a vertex between every pair of adjacent vertices of the cycle. ijpam.eu

2 2. Very Strongly Perfect Graph (VSP) A graph G is said to be very strongly perfect if for every induced sub graph H, Every vertex of H belongs to an independent set of H meeting all maximal cliques of H. Every Even Cycle, C 2n is very strongly perfect and every Odd Cycle, C 2n+1 is non-very strongly perfect. Example 1 Fig.1: Very Strongly Perfect Graph Here, {1} is a independent set which meets all the maximal cliques of graph G. Example 2 Fig.2: Non-Very Strongly Perfect Graph Here,{v 4, v 5, v 6 } is a independent set which does not meet all maximal cliques. Theorem 1. Every even cycle is very strongly perfect. Theorem 2. Evey odd cycle is non very strongly perfect for n > 3. ijpam.eu

3 3. Bipartite Graph A bipartite graph is a graph whose vertex set V can be partitioned into two disjoint sets V 1 and V 2 such that every edge e = uv has one end in V 1 and other in V 2. Fig.3: Bipartite Graph Theorem 3. Every bipartite graph is very strongly perfect. Proof. Let G be a bipartite graph. Every bipartite graph G(V, E) contains no odd cycle. As every path and even cycle are very strongly perfect. Therefore every bipartite graph is very strongly perfect. Hence the proof. Theorem 4. Independence number of bipartite graph is max { V 1, V 2 } Proof. Let G be a bipartite graph. Therefore vertex set V of graph G(V, E) can be partitioned into two disjoint set of vertices V 1 and V 2, Such that V 1 = m and V 2 = n. Case 1 : If m > n. Then I = V 1 gives the independent set with the maximum number of vertices which meet all maximal cliques of size two i.e. k 2. Therefore I = V 1 = m. Case 2 : If m < n. Then I = V 2 gives the independent set with the maximum number of vertices which meet all maximal cliques of size two i.e. k 2. Therefore I = V 2 = n. Hence from case1 and case 2 we get, Independence number Ψ(G) = max{ V 1, V 2 } ijpam.eu

4 Observation 5. Every bipartite graph G contains maximal clique of size two only. Proof. We prove result by contradiction. Suppose a bipartite graph G contain a maximal clique k n, n > 2. As k n is a complete graph of n vertices, therefore each vertex v i in k n adjacent to all the remaining vertices of k n, which gives there exist an edge e = v i v j of the graph G, such that e lies in V 1 or in V 2. This gives contradiction to the fact that G is bipartite. Hence our assumption is wrong. Therefore every bipartite graph G contains a maximal clique of size two. 4. Star graph The Star graph S k is the complete bipartite graph k 1,n. It is a tree with one internal node and n leaves. Fig.4: Star graph Independent set = {v 1 } or {v 2, v 3, v 4, v 5 } which meets all maximal cliques. Theorem 6: Every star graph S k, is very strongly perfect. Proof. Let G be a star graph S k. Let I = {v 1 } be the independent set of cardinality one. v 1 is adjacent to all remaining vertices of G. v 1 meets all the maximal cliques of G. G is very strongly perfect. Hence proved. Observations 7. Let G be a star graph S k. Then G has a minimal dominating set of maximal cardinality k. Proof. Let G be a star graph S k. G has k maximal cliques on two vertices. Since G has k-copies of k 2, there exist a minimal dominating set of cardinality k, which meets all k 2. G has minimal dominating set of maximum cardinality k. Hence proved. ijpam.eu

5 5. Cubic Bipartite Graph Every Bipartite graph contains no odd cycle therefore every bipartite graph is very strongly perfect. A bipartite graph G is said to be cubic bipartite graph if each of its vertex is of degree three. Let G be a cubic bipartite graph and n be the number of vertices. We know that minimum number of vertices in cubic bipartite graph equals to six. Also there is no graph of odd vertices. The number of vertices in the largest independent set of graph G called independence number is denoted by Ψ(G). For n = 6. Fig.5: Cubic Bipartite Graph The largest independent set = {v 1, v 2, v 3 } also Ψ(G) = n/2 = 3. Summarizing the above examples we get the following results. Theorem 8: Every cubic bipartite graph is very strongly perfect graph. Proof. Consider a cubic bipartite graph G of n vertices, as the vertex set V can be partitioned into two sets V 1 and V 2 such that V = V 1 V 2 and V 1 V 2 = φ with V 1 = V 2 = n/2. Therefore cardinality of largest independent set is n/2. Also each vertex belongs to an independent set meet the maximal clique k 2. Hence graph G is very strongly perfect. Observation 9. For a cubic bipartite graph, Independence number Ψ(G) equals to n/2, where n is number of vertices. Proof. Let G be a cubic bipartite graph with n vertices. As the vertex set V can be partitioned into two sets V 1 and V 2 such that V = V 1 V 2 and V 1 V 2 = φ with V 1 = V 2 = n/2. That means the largest independent set is either V 1 or V 2 not both. Therefore cardinality of largest independent set is equals to n/2. Hence, Independence number Ψ(G) equals to n/2. ijpam.eu

6 6. Wheel Graph A wheel graph W m, m 3 is a graph obtained by joining all vertices of cycle C m to a further vertex c called centre. Fig.6: Wheel Graph Theorem 10: Every wheel graph W 2m is very strongly perfect. Proof. Let G(V, E) be a wheel graph. Since, G constructed by joining all vertices of cycle C 2m to a further vertex C. Therfore for every induced sub graph H, every vertex of H belongs to its independent set meets all maximal cliques of size three i.e. k 3. Hence G is a very strongly perfect graph. Theorem 11: Every wheel graph W 2m+1 is non very strongly perfect. Proof. Let G be a wheel graph. Therefore G is constructed by joining all vertices of cycle C 2m+1 to a further vertex c called center. Therefore C n = {u 1 u 2 u 3 u 2m+1 u 1 }. Let I = {u i /i = 1, 3, 5,, 2 m+1 }, i.e. I be the independent set containing all odd vertices. Hence, for all k (1, 2, 3,, 2 m+1 ) we get, u k I if k = odd. u k / I if k = even. As 2m + 1 is odd number, u 2m+1 I. Therefore, there exist an edge e = u 2m+1 u 1 in G(V, E) belongs to an independent set I. That proves, I contain two adjacent vertices. Which gives contradiction to the fact that I is an independent set. Hence G is non very strongly perfect graph. ijpam.eu

7 Observation 12. Independence number of wheel graph W 2m is m. Proof. Let G be a wheel graph of 2m + 1 vertices. As a wheel graph G contains a cycle C 2m which is of maximum length 2m. That means C 2m = {u 1 u 2 u 3 u 2m u 1 }. Therefore every vertex belongs to cycle C 2m meet all cliques of size three i.e. k 3. Set I = {u i /i = even or odd} be the set of all alternating vertices from cycle C 2m. Clearly I is a maximum independent set meet all maximal cliques of size three. Which gives cardinality of I is half the length of cycle C 2m i.e. I = m. Hence, independence number Ψ(G) equals to m. 7. Gear Graph A gear graph G is obtained from wheel graph by adding a vertex between every pair of adjacent vertices of the cycle. The gear graph G n has (2n + 1) vertices and 3n edges. Fig.7: Gear Graph Theorem 13: Every gear graph G n is very strongly perfect. Proof. Let G be a gear graph which is obtained by wheel graph, by adding vertex between every pair of adjacent vertices in the cycle. We have V = 2n + 1 and E(G) = {c u i /i = odd} {u i u i+1 /1 i 2n 1} {u 2n u 1 }. Therefore there exists a maximal clique of size two. Let, I = {u j /j = odd}, as j is odd every vertex in I adjacent to centre C of graph G. i.e every vertex in I meet all maximal cliques of size two. Hence every gear graph G is very strongly perfect. ijpam.eu

8 Observation 14. Independence number of gear graph is n. Proof. Let G be a gear graph. There exist a cycle of length 2n, which is maximum. Therefore every alternating vertex in cycle C 2n forms a maximal independent set of size n. Which gives independence number Ψ(G) of gear graph is n. Hence the proof. Conclusion We have analysed the structure of very strongly perfect graph in bipartite graphs, cubic bipartite graphs, wheel graph and gear graphs. We have presented the characterisation of very strongly perfect graph in bipartite graphs, cubic bipartite graphs, wheel graph and gear graphs. In future these investigations will be extended to the remaining well known architecture. References [1] J.A. Bondy and U.S.R.Murty, Graph theory with applications., McMillan, London,976 [2] J.A. Bondy and U.S.R.Murty, Graph theory and applications, (North Holand) New York (1976) [3] K.Vaithilingam, Diffrence lebelling of some graph families, International journal of mathematical and statistics invention (IJMSI), E-ISSN: P-ISSN: [4] K Kavitha & N.G. David, Dominator colouring of some classes of graphs, International journal of mathematical archieves ISSN , 3 (11),2012, [5] Erodos, Paul, Renyi, Alfred. Sos, T. Vera, On a problem of graph theory, Studia Sci. Math Hungar., 1 (1966), [6] J.A. Gallian, Dynamic Survey- Graph Labeling, Electronic J.Combinatorics, DS6 (2007), [7] R.Mary Jeya Jothi, A. Amutha, n-colourable Super Strongly Perfect Graphs, Indo- Solvenia Conference on Graph Theory and Applications (Indo-Solv-2013), Kerla(2013). [8] U.S.R. Murthy, Open problems, Trends in mathematics,25(2006) [9] R. Mary Jeya Jothi, A. Amutha, SSP Structure of Some Graph Classes International Journal of pure and Applied Mathematics vol101 No. 6(2015), ijpam.eu

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