A charming class of perfectly orderable graphs (original) (raw)
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A note on perfectly orderable graphs
Discrete Applied Mathematics, 1996
We introduce a new class of perfectly orderable graphs that contains complements of chordal bipartite graphs, unions of two threshold graphs. graphs with Dilworth number at most three. and complements of triangulated graphs.
Some classes of perfectly orderable graphs
Journal of Graph Theory, 1989
In 1981, Chvátal defined the class of perfectly orderable graphs. This class of perfect graphs contains the comparability graphs and the triangulated graphs. In this paper, we introduce four classes of perfectly orderable graphs, including natural generalizations of the comparability and triangulated graphs. We provide recognition algorithms for these four classes. We also discuss how to solve the clique, clique cover, coloring, and stable set problems for these classes.
New properties of perfectly orderable graphs and strongly perfect graphs
Discrete Mathematics, 1991
Hoang, C.T., F. Maffray and M. Preissmann, New properties of perfectly orderable graphs and strongly perfect graphs, Discrete Mathematics 98 (1991) 161-174. We establish a property of minimal nonperfectly orderable graphs, and use this property to generate a class of perfectly orderable graphs which strictly contains all brittle graphs. This class is characterized by the existence, in each induced subgraph, of a vertex which is either the endpoint of no Ps, or the midpoint of no P4, or the mid-point of exactly one P4 and the endpoint of exactly one P4. As a consequence, we show that the number of P4's in a minimal nonperfectly orderable graph is at least in, where n is the number of vertices of the graph. Similar results are obtained for strongly perfect graphs.
A Remark on the Characterization of Triangulated Graphs
Open Journal of Discrete Mathematics
In this study we consider the problem of triangulated graphs. Precisely we give a necessary and sufficient condition for a graph to be triangulated. This give an alternative characterization of triangulated graphs. Our method is based on the so called perfectly nested sequences.
On the complexity of recognizing a class of perfectly orderable graphs
Discrete Applied Mathematics, 1996
Recently Middendorf and Pfeiffer proved that recognizing perfectly orderable graphs is NP-complete. Hoang and Reed had previously studied six natural subclasses of perfectly orderable graphs that are defined by the orientations of the P,'s. Four of the six classes can be recognized in polynomial time. We show in this paper that it is NP-complete to recognize the fifth class.
Triangular embeddings of complete graphs from graceful labellings of paths
Journal of Combinatorial Theory, 2007
We show that to each graceful labelling of a path on 2s + 1 vertices, s 2, there corresponds a current assignment on a 3-valent graph which generates at least 2 2s cyclic oriented triangular embeddings of a complete graph on 12s + 7 vertices. We also show that in this correspondence, two distinct graceful labellings never give isomorphic oriented embeddings. Since the number of graceful labellings of paths on 2s + 1 vertices grows asymptotically at least as fast as (5/3) 2s , this method gives at least 11 s distinct cyclic oriented triangular embedding of a complete graph of order 12s + 7 for all sufficiently large s.
Mathematical and Computer Modelling, 2011
A graph G is k-ordered if for any sequence of k distinct vertices v 1 , v 2 ,. .. , v k of G there exists a cycle in G containing these k vertices in the specified order. In 1997, Ng and Schultz posed the question of the existence of 4-ordered 3-regular graphs other than the complete graph K 4 and the complete bipartite graph K 3,3. In 2008, Meszaros solved the question by proving that the Petersen graph and the Heawood graph are 4-ordered 3-regular graphs. Moreover, the generalized Honeycomb torus GHT(3, n, 1) is 4-ordered for any even integer n with n ≥ 8. Up to now, all the known 4-ordered 3-regular graphs are vertex transitive. Among these graphs, there are only two non-bipartite graphs, namely the complete graph K 4 and the Petersen graph. In this paper, we prove that there exists a bipartite non-vertex-transitive 4-ordered 3-regular graph of order n for any sufficiently large even integer n. Moreover, there exists a non-bipartite non-vertex-transitive 4-ordered 3-regular graph of order n for any sufficiently large even integer n.
Scenic graphs II: non-traceable graphs
1999
A path of a graph is maximal if it is not a proper subpath of any other path of the graph. A graph is scenic if every maximal path of the graph is a maximum length path. In [4] we give a new proof of C. Thomassen's result characterizing all scenic graphs with Hamiltonian path. Using similar methods here we determine all scenic graphs with no Hamiltonian path.
Discrete Mathematics, 2000
Consider the following total order: order the vertices by repeatedly removing a vertex of minimum degree in the subgraph of vertices not yet chosen and placing it after all the remaining vertices but before all the vertices already removed. For which graphs the greedy algorithm on this order gives an optimum vertex-coloring? Markossian, Gasparian and Reed introduced the class of ÿ-perfect graphs. These graphs admit such a greedy vertex-coloring algorithm. The recognition of ÿ-perfect graphs is open. We deÿne a subclass of ÿ-perfect graphs, that can be recognized in polynomial time, by considering the class of graphs with no even hole, no short-chorded cycle on six vertices, and no diamond. In particular, we make use of the following properties: no minimal ÿ-imperfect graph contains a simplicial vertex, a minimal ÿ-imperfect graph which is not an even hole contains no vertex of degree 2.