Showing papers on "Path graph published in 1978"

A recognition algorithm for the total graphs

Abstract: A graph H is called total if there exists a graph G such that there is a one-to-one correspondence between the vertices of H and the vertices and edges of G such that two vertices of H are adjacent iff the corresponding elements of G are adjacent or incident. In this paper we present a linear time algorithm for the recognition of the total graphs. The algorithm is based on the breadth-first search technique.

A lower bound on the number of vertices of a graph

Abstract: In this note, we derive a lower bound for the number of vertices of a graph in terms of its diameter, d, connectivity k and minimum degree p which is sharper than that of Watkins [1] by an amount 2(p k). Let G be any finite, undirected graph with neither loops nor multiple edges. Let n, p, k and d denote the number of vertices, minimum degree, connectivity and diameter of G respectively. Watkins [1] has proved that if k > 1, then n > k(d 1) + 2. He has used Menger's theorem to obtain the above result. In this note we prove a theorem from which Watkins' result follows as a corollary. Our proof is simple and elementary. Moreover the lower bound we obtain is sharper than that of Watkins by the amount 2(p k). THEOREM 1. If k > 1, then k(d-3)+2p+2, if d > 3, n > p + 2, if d = 2, 2, if d= 1. PROOF. Let a and b be two vertices of G at a distance d. Let Ai = {x E V(G)I8(a, x) = i}, i = 0, 1, . . . , d, where 8(a, x) denotes the length of the shortest path between a and x. Clearly, Ai n Ai = 0 for i =# j, A0 = {a) and b E Ad. Let d > 3. If we delete all the vertices in any Ai, 1 S i k for 1 p + 1 and IAd-lI + lAdI > p + 1. Hence n > k(d 3) + 2p + 2. If d = 2, then JAII > p and so n > p + 2. If d = 1, then clearly n > 2. Hence the theorem is proved. Since p > k, we have Watkins' result as a direct corollary to the above theorem. Now we give below examples of two classes of graphs to show that the bounds in our theorem are 'best possible'. EXAMPLE 1 (WATKINS [1]). Let HI, . .. , Hd-l represent disjoint copies of Kk, G is formed as follows: Join each vertex of Hi to each vertex of Hi,+ by an edge (i = 1, . .. , d 2); then join a new vertex u to each vertex of H1 by an edge and similarly join a vertex v to each vertex of Hd 1. The resulting Received by the editors August 1, 1977 and, in revised form, November 23, 1977. AMS (MOS) subject classifications (1970). Primary 05C35. ? American Mathematical Society 1978 211 This content downloaded from 157.55.39.255 on Mon, 01 Aug 2016 05:56:28 UTC All use subject to http://about.jstor.org/terms 212 V. G. KANE AND S. P. MOHANTY graph clearly satisfies the bounds in the theorem but does not show the sharpness of our bound as we have here p = k. EXAMPLE 2. Let d and m be integers at least 2, and let G be the lexi- cographic product of the 2d-circuit with the complete graph Km. Then we have k = 2m, p = 3m 1 and n = 2md. Now G has diameter d and substi- tution yields k(d3) + 2p + 2 = n. In this example we have p > k and hence our lower bound is sharper than that of Watkins by an amount 2(p k). ACKNOWLEDGEMENT. We are extremely thankful to the referee for sugges- ting the class of graphs given in Example 2. REFERENCES 1. M. E. Watkins, A lower bound on the number of vertices of a graph, Amer. Math. Monthly 74 (1967), 297. ELECTRONICS COMMSSION, NEW DELHI, INDIA DEPARTMENT OF MATHEMATICS, INDLAN INSTITUTE OF TECHNOLOGY, KANPUR 208016 INDIA This content downloaded from 157.55.39.255 on Mon, 01 Aug 2016 05:56:28 UTC All use subject to http://about.jstor.org/terms

An upper bound on the size of a largest clique in a graph

Abstract: A graph is point determining if distinct vertices have distinct neighborhoods. The nucleus of a point-determining graph is the set GO of all vertices, v, such that G–v is point determining. In this paper we show that the size, ω(G), of a maximum clique in G satisfies ω(G) ⩽ 2|π (G)O|, where π(G) (the point determinant of G) is obtained from G by identifying vertices which have the same neighborhood.

Graph Decompositions Satisfying Extremal Degree Constraints

Abstract: We show that the vertex set of any graph G with p⩾2 vertices can be partitioned into non-empty sets V1, V2, such that the maximum degree of the induced subgraph 〈Vi〉 does not exceed where pi = |Vi|, for i=1, 2. Furthermore, the structure of the induced subgraphs is investigated in the extreme case.

A new method finding the k-th best path in a graph

Abstract: This paper presents a new algorithm determining the K·th best path without any circuit between two specified vertices in a connected, simple and nonoriented graph.. The method presented here is based on the well·known fact that the minimum set of ring sum of several Euler graphs and a special path between two vertices consists of all paths between the vertices. Lastly, an illustrative example iI given and the efficiency of the algorithm is estimated approximately.

Spanning Forest of a Graph (SPANFO)

Abstract: This chapter discusses an algorithm SPANFO for spanning forest of a graph. One of the most fundamental questions about a graph concerns its connectivity. A graph G is connected if whenever u and v are distinct vertices of G , there is a path in G joining u and v . Even if G is not connected, an equivalence relation can be defined on its vertex set: u and v are related if there is a path between them. A connected component of G is an equivalence class T of vertices of G , under this equivalence relation, together with all of the edges of G which are incident with some vertex of T . The chapter discusses on-line algorithm and off-line algorithms. In an off-line algorithm, the user is presented with a list of all of the edges of G before any computation is done. The algorithm is, therefore, permitted to operate on the entire graph as a unit. This chapter describes two algorithms, namely, a depth-first-search algorithm and the breadth first algorithm offering economies of array space. It discusses a FORTRAN program for the latter algorithm.