
On the semiproper orientations of graphs
A semiproper orientation of a given graph G is a function (D,w) that a...
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Proper Orientation Number of Trianglefree Bridgeless Outerplanar Graphs
An orientation of G is a digraph obtained from G by replacing each edge ...
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Weighted proper orientations of trees and graphs of bounded treewidth
Given a simple graph G, a weight function w:E(G)→N∖{0}, and an orientati...
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Eliminating Odd Cycles by Removing a Matching
We study the problem of determining whether a given graph G=(V, E) admit...
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Generalising the achromatic number to Zaslavsky's colourings of signed graphs
The chromatic number, which refers to the minimum number of colours requ...
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The SingleFace Ideal Orientation Problem in Planar Graphs
We consider the ideal orientation problem in planar graphs. In this prob...
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Duality pairs and homomorphisms to oriented and unoriented cycles
In the homomorphism order of digraphs, a duality pair is an ordered pair...
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On the proper orientation number of chordal graphs
An orientation D of a graph G=(V,E) is a digraph obtained from G by replacing each edge by exactly one of the two possible arcs with the same end vertices. For each v ∈ V(G), the indegree of v in D, denoted by d^_D(v), is the number of arcs with head v in D. An orientation D of G is proper if d^_D(u)≠ d^_D(v), for all uv∈ E(G). An orientation with maximum indegree at most k is called a korientation. The proper orientation number of G, denoted by χ(G), is the minimum integer k such that G admits a proper korientation. We prove that determining whether χ(G) ≤ k is NPcomplete for chordal graphs of bounded diameter, but can be solved in lineartime in the subclass of quasithreshold graphs. When parameterizing by k, we argue that this problem is FPT for chordal graphs and argue that no polynomial kernel exists, unless NP⊆ coNP/ poly. We present a better kernel to the subclass of split graphs and a linear kernel to the class of cobipartite graphs. Concerning bounds, we prove tight upper bounds for subclasses of block graphs. We also present new families of trees having proper orientation number at most 2 and at most 3. Actually, we prove a general bound stating that any graph G having no adjacent vertices of degree at least c+1 have proper orientation number at most c. This implies new classes of (outer)planar graphs with bounded proper orientation number. We also prove that maximal outerplanar graphs G whose weakdual is a path satisfy χ(G)≤ 13. Finally, we present simple bounds to the classes of chordal clawfree graphs and cographs.
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