Some Classical Problems on K1,3-free Split Graphs

Abstract

A set D$\subseteq$V of a graph G is called a total outer-connected dominating set of G if D is a total dominating set of G and G[V$\backslash$D] is connected. The Minimum Total Outer-connected Domination (MTOCD) problem is NP-complete in general graphs, chordal graphs and split graphs. Hence we look into MTOCD problem and found that in $K_{1,3}-$free split graphs the MTOCD problem is polynomial-time solvable. We present an polynomial-time algorithm which computes minimum total outer-connected dominating set in $K_{1,3}-$free split graphs since the problem is NP-complete in $K_{1,5}-$free split graphs as we can observe that from the NP-completeness reduction of split graphs has $K_{1,5}$ as its induced subgraph.The Longest path problem is the problem of finding a simple maximum length path in a graph. Longest path is NP-complete for all graph classes in which Hamiltonian path problem is NP-Complete. Since Hamiltonian path problem is known to be NP-Complete in $K_{1,5}$-free split graphs, it is obvious that in $K_{1,5}$-free split graphs longest path problem is also NP-complete. We present a polynomial-time algorithm to find a longest path in $K_{1,3}$-free split graphs.The decision version of Maximum-Cut problem is known to be NP-complete in 2-split graphs. We propose a polynomial-time algorithm to solve the Maximum-Cut problem in $K_{1,3}$-free split graphs with $d(v)\leq 2, \forall v\in I$, which are a subclass of 2-split graphs. We also present polynomial-time algorithms for the Steiner Path problem, the dominating set problem the Steiner cycle problem and cutwidth problem restricted to $K_{1,3}$-free split graphs.