Perfect graph

A bipartite graph (left) and its line graph (right). The shaded cliques in the line graph correspond to the vertices of the underlying bipartite graph, and have size equal to the degree of the corresponding vertex.

A bipartite graph is a graph that can be colored with two colors, or equivalently a graph that has no odd cycles. Because they have no odd cycles, they have no cliques of three or more vertices, so (when they have at least one edge) their chromatic number and clique number are both two. An induced subgraph of a bipartite graph remains bipartite, with the same equality between chromatic number and clique number, so the bipartite graphs are perfect.[9] They include as subclasses the trees, even-length cycle graphs, lattice graphs, knight's graphs, modular graphs, median graphs, and partial cubes, among many others.[14]

The perfect graph theorem, applied to bipartite graphs, produces a less-obvious equality: in these graphs, the size of the maximum independent set equals the size of the smallest clique cover. Because bipartite graphs are triangle-free, their minimum clique covers consist of a maximum matching (a set of disjoint edges, as many as possible) together with one-vertex cliques for all remaining vertices. And in any graph, the maximum independent set is complementary to a vertex cover, a set of vertices that touches all edges. Therefore, the equality of the maximum independent set and minimum clique cover can be expressed equivalently as an equality between the size of the maximum matching and the minimum vertex cover, the usual formulation of Kőnig's theorem.[15][16]

The same equality can be expressed in another way, involving a different class of perfect graphs. A matching, in any graph ${\displaystyle G}$, is the same thing as an independent set in the line graph ${\displaystyle L(G)}$, a graph that has a vertex for each edge in ${\displaystyle G}$. There is an edge between two vertices in ${\displaystyle L(G)}$ whenever the corresponding two edges in ${\displaystyle G}$ share an endpoint. There are two kinds of cliques in a line graph ${\displaystyle L(G)}$: the ones coming from sets of edges in ${\displaystyle G}$ that all have a common endpoint, and the ones coming from triangles in ${\displaystyle G}$. However, if ${\displaystyle G}$ is bipartite, only the first kind of clique can exist in ${\displaystyle L(G)}$. Therefore, in this case, a clique cover in ${\displaystyle L(G)}$ corresponds to a vertex cover in ${\displaystyle G}$, so in line graphs of bipartite graphs the independence number and clique cover number are equal. Induced subgraphs of line graphs of bipartite graphs remain in the same family (they correspond to the subgraphs of the underlying bipartite graph), and so the line graphs of bipartite graphs are perfect.[16] In particular this is true of the rook's graphs, the line graphs of complete bipartite graphs. Every line graph of a bipartite graph is an induced subgraph of a rook's graph.[17]

Because line graphs of bipartite graphs are perfect, their clique number equals their chromatic number. The clique number of the line graph of a bipartite graph is the maximum degree of any vertex of the underlying bipartite graph. The chromatic number of the line graph of a bipartite graph is the chromatic index of the underlying bipartite graph. It is the minimum number of colors needed to color the edges so that no two edges of the same color share an endpoint. Each color class forms a matching, so the chromatic index is also the minimum number of matchings needed to cover all of the edges of the graph. The equality of maximum degree and chromatic index, in bipartite graphs, is another theorem of Dénes Kőnig.[18] In arbitrary simple graphs, they can differ by one; this is Vizing's theorem.[16]

A line perfect graph, with black bipartite biconnected components, blue ${\displaystyle K_{4}}$, and red triangular books

If a line graph ${\displaystyle L(G)}$ is perfect, the underlying graph ${\displaystyle G}$ is said to be a line perfect graph. The line perfect graphs are the graphs whose biconnected components are bipartite graphs, the complete graph ${\displaystyle K_{4}}$, and triangular books, sets of triangles sharing an edge. Because these components are themselves perfect, and are combined in a way that preserves the property of being perfect, every line perfect graph is itself a perfect graph.[16]

The bipartite graphs, their complements, and the line graphs of bipartite graphs and their complements form four basic classes of perfect graphs that play a key role in the proof of the strong perfect graph theorem. According to the structural decomposition of perfect graphs used as part of this proof, every perfect graph that is not already in one of these four classes can be decomposed by partitioning its vertices into subsets, in one of four ways, called a 2-join, the complement of a 2-join, a homogeneous pair, or a skew partition.[19]

The Hasse diagram of a partially ordered set, and its comparability graph

A partially ordered set is defined by its set of elements, and a comparison relation ${\displaystyle \leq }$ on these elements, obeying the requirements that ≤ is a reflexive relation (for all elements ${\displaystyle x}$, ${\displaystyle x\leq x}$), an antisymmetric relation (if ${\displaystyle x\leq y}$ and ${\displaystyle y\leq x}$, then ${\displaystyle x=y}$, and a transitive relation (if ${\displaystyle x\leq y}$ and ${\displaystyle y\leq z}$, then ${\displaystyle x\leq z}$). Two distinct elements ${\displaystyle x}$ and ${\displaystyle y}$ are said to be comparable if ${\displaystyle x\leq y}$ or ${\displaystyle y\leq x}$, and incomparable otherwise. For instance, the set inclusion operation ${\displaystyle \subseteq }$ is the comparison relation of a partial order on any family of sets. The comparability graph of a partially ordered set is a graph with the set elements as its vertices, and with an edge connecting two vertices whenever those two elements are comparable. Its complement is called an incomparability graph. The comparability graph is not enough to uniquely identify a partial order; different partial orders may have the same comparability graph. For instance, reversing all comparison relations between pairs of elements changes their order but does not change their comparability graph.[20]

Finite comparability graphs (and their complementary incomparability graphs) are always perfect.[21] A clique, in a comparability graph, comes from a subset of elements that are all pairwise comparable; such a subset is called a chain, and it is linearly ordered by the given partial order. An independent set comes from a subset of elements no two of which are comparable; such a subset is called an antichain. Thus, a coloring of a comparability graph is a partition of its elements into antichains, and a clique cover is a partition of its elements into chains. Dilworth's theorem, in the theory of partial orders, states that for every finite partial order, the size of the largest antichain equals the minimum number of chains into which the elements can be partitioned. In the language of graphs, this can be stated as: every finite comparability graph is perfect. Similarly, Mirsky's theorem states that for every finite partial order, the size of the largest chain equals the minimum number of antichains into which the elements can be partitioned, or that every finite incomparability graph is perfect. These two theorems are equivalent via the perfect graph theorem, but Mirsky's theorem is easier than Dilworth's theorem to prove directly: if each element is labeled by the size of the largest chain in which it is maximal, then the subsets with equal labels form a partition into antichains, with the number of antichains equal to the size of the largest chain overall.[22]

Every bipartite graph is the comparability graph of a partial order. If a bipartite graph has its vertices colored, say red and blue, then it is the comparability graph for a comparison relation in which every red vertex is ${\displaystyle \leq }$ every blue vertex, every vertex is ${\displaystyle \leq }$ itself, and there are no other comparable pairs. In the partial orders constructed in this way, the largest chains consist of one red and one blue vertex. Thus, by connecting them through the theory of perfect graphs, Kőnig's theorem can be seen as a special case of Dilworth's theorem.[23]

The permutation graph of the permutation (4,3,5,1,2) connects pairs of elements whose ordering is reversed by the permutation.

A permutation graph is defined from a permutation on a totally ordered sequence of elements (conventionally, the integers from ${\displaystyle 1}$ to ${\displaystyle n}$), which form the vertices of the graph. The edges of a permutation graph connect pairs of elements whose ordering is reversed by the given permutation. These are naturally incomparability graphs, for a partial order in which ${\displaystyle x\leq y}$ whenever ${\displaystyle x}$ occurs before ${\displaystyle y}$ in both the given sequence and its permutation. The complement of a permutation graph is another permutation graph, for the reverse of the given permutation. Therefore, as well as being incomparability graphs, permutation graphs are comparability graphs. In fact, the permutation graphs are exactly the graphs that are both comparability and incomparability graphs.[24] A clique, in a permutation graph, is a subsequence of elements that appear in increasing order in the given permutation, and an independent set is a subsequence of elements that appear in decreasing order. In any perfect graph, the product of the clique number and independence number are at least the number of vertices; the special case of this inequality for permutation graphs is the Erdős–Szekeres theorem.[22]

An interval graph and the intervals defining it

The interval graphs are the incomparability graphs of interval orders, orderings defined by sets of intervals on the real line with ${\displaystyle x\leq y}$ whenever interval ${\displaystyle x}$ is completely to the left of interval ${\displaystyle y}$. In the corresponding interval graph, there is an edge from ${\displaystyle x}$ to ${\displaystyle y}$ whenever the two intervals have a point in common. Coloring these graphs can be used to model problems of assigning resources to tasks (such as classrooms to classes) with intervals describing the scheduled time of each task.[25] Both interval graphs and permutation graphs are generalized by the trapezoid graphs, the incomparability graphs of orderings defined in the same way for a system of trapezoids between two parallel lines, with ${\displaystyle x\leq y}$ whenever trapezoid ${\displaystyle x}$ is completely to the left of trapezoid ${\displaystyle y}$.[26] Systems of intervals in which no two are nested produce a more restricted class of graphs, the indifference graphs, the incomparability graphs of semiorders. These have been used to model human preferences under the assumption that, when items have utilities that are very close to each other, they will be incomparable.[27] Another special case of the interval graphs is the family of trivially perfect graphs, formed from intervals where each two are either nested or disjoint.[28] These are the comparability graphs of ordered trees, and the graphs for which, in every induced subgraph, the independence number equals the number of maximal cliques.[29]

A split graph

A split graph is a graph that can be partitioned into a clique and an independent set. When the clique is chosen to be maximal, such a graph can be colored optimally by assigning one color to each clique vertex, and assigning each independent set vertex the color of a non-adjacent clique vertex. Therefore, these graphs have equal clique numbers and chromatic numbers, and are perfect.

A little more generally, the unipolar graphs are the graphs that can be partitioned into a clique and a cluster graph, a disjoint union of cliques. These include, for instance, the complements of bipartite graphs, as they can be partitioned into two cliques. The unipolar graphs and their complements together form the class of generalized split graphs. Almost all perfect graphs are generalized split graphs, in the sense that the fraction of perfect ${\displaystyle n}$-vertex graphs that are generalized split graphs goes to one in the limit as ${\displaystyle n}$ grows arbitrarily large.[30]

It follows from this result that other limiting properties of almost all perfect graphs can be determined by studying the generalized split graphs. In this way, it has been shown that almost all perfect graphs contain a Hamiltonian cycle. If ${\displaystyle H}$ is an arbitrary graph, the limiting probability that ${\displaystyle H}$ occurs as an induced subgraph of a large random perfect graph is 0, 1/2, or 1, respectively as ${\displaystyle H}$ is not a generalized split graph, is unipolar or co-unipolar but not both, or is both unipolar and co-unipolar.[31]

Several families of perfect graphs can be characterized by an incremental construction in which the graphs in the family are built up by adding one vertex at a time, according to certain rules, which guarantee that after each vertex is added the graph remains perfect.

• The chordal graphs are the graphs formed by a construction of this type in which, at the time a vertex is added, its neighbors form a clique. The maximum clique in the graph is the largest clique obtained in this way from a newly added vertex and its neighbors. Chordal graphs may also be characterized as the graphs that have no holes (even or odd). They include as special cases the forests, the interval graphs, split graphs, and the maximal outerplanar graphs. The k-trees, central to the definition of treewidth, are chordal graphs in which all maximal cliques have the same size.

Three types of vertex addition in a distance-hereditary graph

• The distance-hereditary graphs are formed, starting from a single-vertex graph, by repeatedly adding degree-one vertices ("pendant vertices") or copies of existing vertices (with the same neighbors). Each time a vertex is copied, the vertex and its copy may be adjacent (forming true twins) or non-adjacent (forming false twins). These graphs have the property that, in every connected induced subgraph, the distances between vertices are the same as in the whole graph. If only the twin operations are used, the result is a cograph.[32] The cographs are the comparability graphs of series-parallel partial orders[33] and can also be formed by a different construction process combining complementation and the disjoint union of graphs.[34]
• The graphs that are both chordal and distance-hereditary are called Ptolemaic graphs, because their distances obey Ptolemy's inequality.[35] They have a restricted form of the distance-hereditary construction sequence, in which a false twin can only be added when its neighbors would form a clique.[32] They include as special cases the windmill graphs consisting of cliques joined at a single vertex, and the block graphs in which each biconnected component is a clique.[35]
• The threshold graphs are formed from an empty graph by repeatedly adding either an isolated vertex (connected to nothing else) or a universal vertex (connected to all other vertices).[36] They are special cases of the split graphs and the trivially perfect graphs. They are exactly the graphs that are both trivially perfect and the complement of a trivially perfect graph; they are also exactly the graphs that are both cographs and split graphs.[37]

If the vertices of a chordal graph are colored in the order of an incremental construction sequence using a greedy coloring algorithm, the result will be an optimal coloring. The reverse of the vertex ordering used in this construction is called an elimination order. Similarly, if the vertices of a distance-hereditary graph are colored in the order of an incremental construction sequence, the resulting coloring will be optimal. If the vertices of a comparability graph are colored in the order of a linear extension of its underlying partial order, the resulting coloring will be optimal. This property is generalized in the family of perfectly orderable graphs, the graphs for which there exists an ordering that, when restricted to any induced subgraph, causes greedy coloring to be optimal. The cographs are exactly the graphs for which all vertex orderings have this property.

Many other classes of perfect graphs have been studied. For instance, the strongly perfect graphs are graphs in which, in every induced subgraph, there exists an independent set that intersects all maximal cliques. The Meyniel graphs have been called very strongly perfect because in them, every vertex belongs to such an independent set. The Meyniel graphs can also be defined as the graphs in which every odd cycle of length five or more has at least two chords.[38]

A parity graph that is neither distance-hereditary nor bipartite

A parity graph is defined by the property that between every two vertices, all induced paths have the same parity: either they are all even in length, or they are all odd in length. This is true, for instance, of the distance-hereditary graphs, in which all induced paths between the same two vertices have the same length.[39] It is also true of the bipartite graphs: in a bipartite graph, if two vertices are on the same side of the bipartition, then all paths between them have even length, and if they are on opposite sides then all paths between them have odd length. The parity graphs are a subclass of the Meyniel graphs: in any long odd cycle of a parity graph, at least two chords are needed to prevent the cycle being partitioned into two induced paths of different parity. The prism over any parity graph (its Cartesian product with a single edge) is another parity graph, and the parity graphs are the only graphs whose prisms are perfect.[40]

Tolerance graphs are defined by specifying both an interval of the real line for each vertex and a non-negative real number, the tolerance of the vertex. Two vertices are adjacent whenever their intervals overlap for a length that is at least as large as the minimum of their two tolerances. Thus, the interval graphs are a special case of tolerance graphs, having tolerance zero. Like interval graphs, tolerance graphs can be used to model scheduling problems, where the tasks to be scheduled may only share resources for a short period of time (the tolerance of the tasks). The complements of tolerance graphs form a subclass of the perfectly orderable graphs.[41]