Abstract

An orientation D of a graph G is a digraph obtained from G by replacing each edge by exactly one of the two possible arcs with the same ends. An orientation D of a graph G is a k-orientation if the in-degree of each vertex in D is at most k. An orientation D of G is proper if any two adjacent vertices have different in-degrees in D. The proper orientation number of a graph G, denoted by →χ (G), is the minimum k such that G has a proper k-orientation. A weighted orientation of a graph G is a pair (D, w), where D is an orientation of G and w is an arc-weighting A(D) → N \ {0}. A semi-proper orientation of G is a weighted orientation (D, w) of G such that for every two adjacent vertices u and v in G, we have that S(d,w)(v) S(d,w)(u), where S(d,w)(v) is the sum of the weights of the arcs in (D, w) with head v. For a positive integer k, a semi-proper k-orientation (D, w) of a graph G is a semi-proper orientation of G such that maxv μV(G) S(d,w)(v) ≤ k. The semi-proper orientation number of a graph G, denoted by →χs(G), is the least k such that G has a semi-proper k-orientation. In this work, we first prove that →χs(G) μ {ω(G) - 1, ω(G)} for every split graph G, and that, given a split graph G, deciding whether →χs(G)=ω(G) - 1 is an NP-complete problem. We also show that, for every k, there exists a (chordal) graph G and a split subgraph H of G such that →χ(G) ≤ k and →χ(H)=2k - 2. In the sequel, we show that, for every n ≥ p(p + 1), →χs(Ppn) =[3/2p], where Ppnis the pthpower of the path on n vertices. We investigate further unit interval graphs with no big clique: we show that →χ(G) ≤ 3 for any unit interval graph G with ω(G) = 3, and present a complete characterization of unit interval graphs with →χ(G)= ω(G) = 3. Then, we show that deciding whether →χs(G)=ω(G) can be solved in polynomial time in the class of co-bipartite graphs. Finally, we prove that computing →χs(G) is FPT when parameterized by the minimum size of a vertex cover in G or by the treewidth of G. We also prove that not only computing →χs(G) but also →χ(G), admits a polynomial kernel when parameterized by the neighbourhood diversity plus the value of the solution. These results imply kernels of size 40(k2)and 0(2kk2), in chordal graphs and split graphs, respectively, for the problem of deciding whether →χs(G) ≤ k parameterized by k. We also present exponential kernels for computing both →χ(G) and →χs(G) parameterized by the value of the solution when G is a cograph. On the other hand, we show that computing →χs(G) does not admit a polynomial kernel parameterized by the value of the solution when G is a chordal graph, unless NP coNP/poly.