For a fixed number of colors, we show that, in node-weighted split graphs, cographs, and graphs of bounded tree-width, one can determine in polynomial time whether a proper list-coloring of the vertices of a graph such that the total weight of vertices of each color equals a given value in each part of a fixed partition of the vertices exists. We also show that this result is tight in some sens...

The complexity of the simple maxcut problem is investigated for several special classes of graphs. It is shown that this problem is NP-complete when restricted to one of the following classes: chordal graphs, undirected path graphs, split graphs, tripartite graphs, and graphs that are the complement of a bipartite graph. The problem can be solved in polynomial time, when restricted to graphs wi...

The Grundy number of a graph G is the largest number of colors used by any execution of the greedy algorithm to color G. The problem of determining the Grundy number of G is polynomial if G is a P 4-free graph and N P-hard if G is a P 5-free graph. In this article, we define a new class of graphs, the fat-extended P 4-laden graphs, and we show a polynomial time algorithm to determine the Grundy...

We present two characterizations of planarity based on Trémaux trees (i.e. DFS trees). From the last one, we deduce a simple and efficient planarity test algorithm which is eventually a new implementation of the Hopcroft Tarjan planarity algorithm. We finally recall a theorem on “cotree critical non-planar graphs” which very much simplify the search of a Kuratowski subdivision in a non-planar g...

In the past, diierent authors developed distinct approaches to the problem of transitive orientation. This also resulted in diierent ideas and diierent theorems which seem unrelated. In this paper we show the connections between these theories and present a new algorithm to recognize a comparability graph. A comparability graph is an undirected graph G = (V; E), jV j = n, jEj = m, in which ever...

A graph G is matching-decyclable if it has a matching M such that G−M is acyclic. Deciding whether G is matching-decyclable is an NPcomplete problem even if G is 2-connected, planar, and subcubic. In this work we present characterizations of matching-decyclability in the following classes: chordal graphs, split graphs, distance-hereditary graphs, cographs, and Hamiltonian subcubic graphs. All t...

A graph is k-choosable if it admits a proper coloring of its vertices for every assignment of k (possibly different) allowed colors to choose from for each vertex. It is NP-hard to decide whether a given graph is k-choosable for k ≥ 3, and this problem is considered strictly harder than the k-coloring problem. Only few positive results are known on input graphs with a given structure. Here, we ...

The problem of dynamically recognizing a graph property calls for e ciently decid ing if an input graph satis es the property under repeated modi cations to its set of vertices and edges The input to the problem consists of a series of modi cations to be performed on the graph The objective is to maintain a representation of the graph as long as the property holds and to detect when it ceases t...

We present a graph algorithm to find fundamental cycles aligned with the principal curvature directions of a surface. Specifically, we use the tree-cotree decomposition of graphs embedded in manifolds, guided with edge weights, in order to produce these cycles. Our algorithm is very quick compared to existing methods, with a worst case running time of O(n logn + gn) where n is the number of fac...

This paper investigates a variant of the general problem of assigning channels to the stations of a wireless network when the graph representing the possible interferences is a matrogenic graph. In our problem, channels assigned to adjacent vertices must be at least two apart, while channels assigned to vertices at distance two must be different. An exact linear time algorithm is provided for t...