Complexity of total outer-connected domination problem in graphs

Abstract

is to find a total outer-connected dominating set of minimum cardinality. The Total Outer-connected Domination Decision problem, the decision version of the Minimum Total Outer-connected Domination problem, is known to be NP-complete for general graphs. In this paper, we strengthen this NP-completeness result by showing that the Total Outer-connected Domination Decision problem remains NP-complete for chordal graphs, split graphs, and doubly chordal graphs. We prove that the Total Outer-connected Domination Decision problem can be solved in linear time for bounded tree-width graphs. We, then, propose a linear time algorithm for computing the total outer-connected domination number, the cardinality of a minimum total outerconnected dominating set, of a tree. We prove that the Minimum Total Outer-connected Domination problem cannot be approximated within a factor of (1 − ε) ln |V | for any ε > 0, unless NP ⊆ DTIME (|V | O(log log |V |) ). Finally, we show that the Minimum Total Outerconnected Domination problem is APX-complete for bounded degree graphs.

Hamiltonian Path in $$K_{1,t}$$-free Split Graphs- A Dichotomy

“…Arti Pandey and B.S. Panda have showed in [23] that the total outer-connected dominating set is NP-complete in split graphs. It is interesting to see that the reduction instances are K 1,5 -free split graphs.…”