Without loss of generality, we assume the following:. For each element e i of e, two fields e i. Thus, the fields e i.
For the graph of Figure 1 , the array e is shown in the Table 1. Table 1. To find the disjoint cliques on interval graphs, we have to first compute all maximal cliques and the time complexity of which given in the following lemma. One more important characterization of the interval graph with respect to cliques is given by Gilmore and Hoffman [ 18 ].
Graph theory examples
It is stated as follows:. A graph G is an interval graph if and only if the maximal cliques of G can be linearly ordered in such a way that for every vertex v of G, the maximal cliques containing v occur consecutively [ 18 ]. Using Lemma-1, we can determine all maximal cliques. If p u is the largest subscript of the maximal cliques in which u belongs, then we call the clique C p u as end clique of u, i. From Lemma-2, we have. As it. Hence, any one among. This means the graph G is disconnected. The array FirstDisjoint plays an important role for construction of the network N.
An algorithm to compute this array is presented below:. Theorem-1 : Algorithm FD can be computed in n 2 time in sequential. Similarly, Step-3 and Step-4 takes n 2 time. Therefore, overall time complexity of the Algorithm FD is of n 2. Hence the theorem. The network N has also a. A path having maximum total weight among all paths from A 0 to A m is called the maximum weight path. Proceedings of the Steklov Institute of Mathematics :S1, Autonomous Agents and Multi-Agent Systems 22 :2, Flows in Networks.
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