Convex obstacle numbers of outerplanar graphs and bipartite permutation graphs

Further results on strong edge-colourings in outerplanar graphs

An edge-colouring is {\em strong} if every colour class is an induced matching. In this work we give a formulae that determines either the optimal or the optimal plus one strong chromatic index of bipartite outerplanar graphs. Further, we give an improved upper bound for any outerplanar graph which is close to optimal. All our proofs yield efficient algorithms to construct such colourings.

Graphs with Plane Outside-Obstacle Representations

An \emph{obstacle representation} of a graph consists of a set of polygonal obstacles and a distinct point for each vertex such that two points see each other if and only if the corresponding vertices are adjacent. Obstacle representations are a recent generalization of classical polygon--vertex visibility graphs, for which the characterization and recognition problems are long-standing open questions. In this paper, we study \emph{plane outside-obstacle representations}, where all obstacles lie in the unbounded face of the representation and no two visibility segments cross. We give a combina…

Eternal vertex cover number of maximal outerplanar graphs

Eternal vertex cover problem is a variant of the classical vertex cover problem modeled as a two player attacker-defender game. Computing eternal vertex cover number of graphs is known to be NP-hard in general and the complexity status of the problem for bipartite graphs is open. There is a quadratic complexity algorithm known for this problem for chordal graphs. Maximal outerplanar graphs forms a subclass of chordal graphs, for which no algorithm of sub-quadratic time complexity is known. In this paper, we obtain a recursive algorithm of linear time for computing eternal vertex cover number o…

在多项式时间内计算外平面图的forcing谱

Computing the forcing spectrum of outerplanar graphs in polynomial time

针对平面图最小forcing数计算复杂度这一开放问题，本文提出首个多项式时间算法，可计算外平面图的所有可能forcing数构成的forcing谱，为该问题的求解迈出重要一步。

Graph drawings with one bend and few slopes

We consider drawings of graphs in the plane in which edges are represented by polygonal paths with at most one bend and the number of different slopes used by all segments of these paths is small. We prove that $\lceil\fracΔ{2}\rceil$ edge slopes suffice for outerplanar drawings of outerplanar graphs with maximum degree $Δ\geq 3$. This matches the obvious lower bound. We also show that $\lceil\fracΔ{2}\rceil+1$ edge slopes suffice for drawings of general graphs, improving on the previous bound of $Δ+1$. Furthermore, we improve previous upper bounds on the number of slopes needed for planar dra…

Defective incidence coloring of graphs

We define the $d$-defective incidence chromatic number of a graph, generalizing the notion of incidence chromatic number, and determine it for some classes of graphs including trees, complete bipartite graphs, complete graphs, and outerplanar graphs. Fast algorithms for constructing the optimal $d$-defective incidence colorings of those graphs are presented.