Orderable Graphs Research Articles

Bipartable graphs

We consider a construction which associated with a graph G another graph G′ such that if G′ is a bipartite graph, G is perfectly orderable. For such a graph G we give a polynomial algorithm for an optimal coloring by delivering a perfect order on its nodes. This class of graphs is shown to be different from the known classes of perfectly orderable graphs.

Perfectly orderable graphs are quasi-parity graphs: a short proof

We show that perfectly orderable graphs are quasi-parity graphs by exhibiting two nodes which are not linked by a chordless odd chain. This proof is short and simpler than the one given by H. Meyniel.

Four classes of perfectly orderable graphs

AbstractA graph is called “perfectly orderable” if its vertices can be ordered in such a way that, for each induced subgraph F, a certain “greedy” coloring heuristic delivers an optimal coloring of F. No polynomial‐time algorithm to recognize these graphs is known. We present four classes of perfectly orderable graphs: Welsh–Powell perfect graphs, Matula perfect graphs, graphs of Dilworth number at most three, and unions of two threshold graphs. Graphs in each of the first three classes are recognizable in a polynomial time. In every graph that belongs to one of the first two classes, we can find a largest clique and an optimal coloring in a linear time.

A New Property of Critical Imperfect Graphs and some Consequences

We prove a new property of critical imperfect graphs. As a consequence, we define a new class of perfect graphs. This class contains perfectly orderable graphs and graphs in which that every odd cycle has two chords.

Generalized neighbourhoods and a class of perfectly orderable graphs

Two classes of perfectly orderable graphs are characterized; they are obtained by considering an ordering of the nodes based on the vicinal preorder and on an extension of it: instead of the set N( u) of neighbours of a node u, we consider the set N ∗(u) of nodes at distance 1 or 2 from u. In the ordering u comes before ν if N ∗(u)∪{u} ⊇ N ∗(ν) and N ∗(ν)∪{v}⊉N ∗(u) . Graphs for which such orderings are perfect are characterizeed in terms of a finite number of forbidden subgraphs. These classes are compared to some other related classes of perfect graphs. Finally one introduces graphs where any 2 nodes are comparable in the preorder defined by inclusion of sets N ∗(u) .

A note on superbrittle graphs

Perfectly orderable graphs are such that an ordering x 1> 2>…> x n of the nodes can be found for which the sequential node coloring algorithm based on this order (“always use the smallest possible color”) gives an optimum coloring for any subgraph. We characterize in terms of forbidden subgraphs two subclasses of perfectly orderable graphs: the superbrittle graphs are such that in any subgraph a node x can never be a midpoint of an induced P 4 (path on 4 nodes) and an endpoint of an induced P 4. Maxibrittle graphs are such that in any subgraph there is no induced P 4 having a node of maximum degree as an endpoint and there is no induced P 4 having a node of minimum degreeas a midpoint.

The Dimension of a Comparability Graph

Dushnik and Miller defined the dimension of a partial order P as the minimum number of linear orders whose intersection is P. Ken Bogart asked if the dimension of a partial order is an invariant of the associated comparability graph. In this paper we answer Bogart’s question in the affirmative. The proof involves a characterization of the class of comparability graphs defined by Aigner and Prins as uniquely partially orderable graphs. Our characterization of uniquely partially orderable graphs is another instance of the frequently encountered phenomenon where the obvious necessary condition is also sufficient.

The dimension of a comparability graph

Dushnik and Miller defined the dimension of a partial order P as the minimum number of linear orders whose intersection is P. Ken Bogart asked if the dimension of a partial order is an invariant of the associated comparability graph. In this paper we answer Bogart's question in the affirmative. The proof involves a characterization of the class of comparability graphs defined by Aigner and Prins as uniquely partially orderable graphs. Our characterization of uniquely partially orderable graphs is another instance of the frequently encountered phenomenon where the obvious necessary condition is also sufficient. 1. Notation and definitions. In this paper we consider a partial order as an irreflexive, transitive binary relation. With a binary relation R on a set A we associate a graph G(R) whose vertex set is A with distinct vertices x and y joined by an edge iff x R y or y R x. A graph G is called a comparability graph if there exists a partial order P for which G = G(P). Aigner and Prins [1] called a comparability graph G a uniquely partially orderable (UPO) graph if G = G(P) = G(Q) implies P = Q or P = Q where Q denotes the dual of Q. Let A' be a graph and let {Gx\x G V(X)} be a family of graphs. Then the (Sabidussi) A-join [9] of this family is the graph with vertex set {(x,.}')| x G V(X), y G V(GX)} with (x,y) adjacent to (z, w) iffx is adjacent to z in X or x = z and y is adjacent to w in Gx. Every graph X is isomorphic to the Adjoin of a family of trivial graphs. If a graph G is isomorphic to the A-join of a family {Gx\x G V(X)} whereX is nontrivial and at least one Gx is nontrivial, then G is said to be decomposable; otherwise G is said to be indecomposable. Let G be a graph and let AT be a subset of V(G). K is said to be partitive iff for every vertex x with x G K, if there exists a vertex y G K such that x and y are adjacent, then x is adjacent to every vertex in K. A partitive subset K is said to be nontrivial when K is not the empty set, a singleton, or the entire vertex set. It is easy to see that a graph is indecomposable iff it has no nontrivial partitive sets. Now let P be a partial order on a set A and let {Qa\a G A) be a family of partial orders. If we denote the set on which each Qa is defined by Aa, then the ordinal product [2] of this family over P is the partial order S on the set {(a,b)\a G A,b G Aa) in which (al,bl)S{a2,bi) iff ax P a2 or ax = a2 and bx Qo| b2. Clearly the comparability graph G(S) is the G(P)-]oin of the family {G(Qa)\a G A). Let e and/be edges of a graph G. Gilmore and Hoffman [6] defined a strong Received by the editors May 12, 1975 and, in revised form, January 20, 1976. AMS (MOS) subject classifications (1970). Primary 06A10, 05C20.

Uniquely Partially Orderable Graphs

Uniquely Partially Orderable Graphs