An optimal algorithm for finding the minimum cardinality dominating set on permutation graphs

Acyclic domination on bipartite permutation graphs

“…Using the result, we prove that a minimum dominating set of G can be transformed into a minimum acyclic dominating set of G in linear time. Finally, we present a linear algorithm for the acyclic domination problem on bipartite permutation graphs, based upon the linear algorithm for minimum domination problem on permutation graphs proposed by Chao et al [2]. As a matter of fact, we show that the acyclic domination problem of bipartite permutation graphs is linear time solvable.…”

“…Since G is a bipartite graph, the induced subgraph D contains no odd-cycle (including 3-cycle). Now assume C h be a cycle with even length h 6 contained in some consecutive BFS-levels of G, then there always exists some i such that |V (C h ) ∩ i+j s=i H s | j + 3 where j ∈ {1, 2}, which contradicting the fact that D satisfies property (2). Therefore, D contain only some 4-cycles.…”

“…Proof. Let G = (S, T , E) be a bipartite permutation graph and D be a minimum dominating set of G which satisfies property (2). Since G is a bipartite graph, the induced subgraph D contains no odd-cycle (including 3-cycle).…”

“…|D ∩ H i+2 | 1 by property (2), then the induced subgraph D ∩ ( j =i+2 j =i−1 H j ) contains no 4-cycles. Therefore, we eliminate the 4-cycle (a, b, c, d) of D and get a new minimum dominating set D which has less cycles than D.…”

“…, H k = {w ∈ V | d G (x, w) = k} be the BFSlevels of G defined as before.Our next result shows that if G has a minimum dominating set D which satisfies the following property, then γ a (G) = γ (G). i∈{0,1,...,k} j ∈{1If G = (S, T , E) is a bipartite permutation graph and G has a minimum dominating set D which satisfies property(2), then γ a (G) = γ (G).…”