A caterpillar graph is a tree which on removal of all its pendant vertices leaves a chordless path. The chordless path is called the backbone of the graph. The edges from the backbone to the pendant vertices are called the hairs of the caterpillar graph. Ortiz and Villanueva (C.Ortiz and M.Villanueva, Discrete Applied Mathematics, 160(3): 259-266, 2012) describe an algorithm, linear in the size of the output, for finding a family of maximal independent sets in a caterpillar graph.In this paper, we propose an algorithm, again linear in the output size, for a generalised caterpillar graph, where at each vertex of the backbone, there can be any number of hairs of length one and at most one hair of length two.
Neighbourly set of a graph is a subset of edges which either share an end point or are joined by an edge of that graph. The maximum cardinality neighbourly set problem is known to be NP-complete for general graphs. Mahdian (M.Mahdian, On the computational complexity of strong edge coloring, Discrete Applied Mathematics, 118:239-248, 2002) proved that it is in polynomial time for quadrilateral-free graphs and proposed an O(n 11 ) algorithm for the same (along with a note that by a straightforward but lengthy argument it can be proved to be solvable in O(n 5 ) running time). In this paper we propose an O(n 2 ) time algorithm for finding a maximum cardinality neighbourly set in a quadrilateral-free graph.
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