cograph is a graph all of whose connected induced subgraphs have diameter at most 2. A cograph is a graph in which every connected component is a distance-hereditary graph with diameter at most 2. A cograph is a graph with clique-width at most 2. A cograph is a comparability graph of a series-parallel partial order. A cograph is a permutation graph of a separable permutation. A cograph is a graph all of whose minimal chordal completions are trivially perfect graphs. A cograph is a hereditarily well-colored graph, a graph such that every greedy coloring of every induced subgraph uses an optimal number of colors. A graph is a cograph if and only if every vertex order of the graph is a perfect order, since having no P4 means that no obstruction to a perfect order will exist in any vertex order. Cotrees A cotree is a tree in which the internal nodes are labeled with the numbers 0 and 1. Every cotree T defines a cograph G having the leaves of T as vertices, and in which the subtree rooted at each node of T corresponds to the induced subgraph in G defined by the set of leaves descending from that node: A subtree consisting of a single leaf node corresponds to an induced subgraph with a single vertex. A subtree rooted at a node labeled 0 corresponds to the union of the subgraphs defined by the children of that node. A subtree rooted at a node labeled 1 corresponds to the join of the subgraphs defined by the children of that node; that is, we form the union and add an edge between every two vertices corresponding to leaves in different subtrees. Alternatively, the join of a set of graphs can be viewed as formed by complementing each graph, forming the union of the complements, and then complementing the resulting union. An equivalent way of describing the cograph formed from a cotree is that two vertices are connected by an edge if and only if the lowest common ancestor of the corresponding leaves is labeled by 1. Conversely, every cograph can be represented in this way by a cotree. If we require the labels on any root-leaf path of this tree to alternate between 0 and 1, this representation is unique. Computational properties Cographs may be recognized in linear time, and a cotree representation constructed, using modular decomposition, partition