Lexicographic breadth-first search

Algorithm

The lexicographic breadth-first search algorithm replaces the queue of vertices of a standard breadth-first search with an ordered sequence of sets of vertices. The sets in the sequence form a partition of the remaining vertices. At each step, a vertex v from the first set in the sequence is removed from that set, and if that removal causes the set to become empty then the set is removed from the sequence. Then, each set in the sequence is replaced by two subsets: the neighbors of v and the non-neighbors of v. The subset of neighbors is placed earlier in the sequence than the subset of non-neighbors. In pseudocode, the algorithm can be expressed as follows:

• Initialize a sequence Σ of sets, to contain a single set containing all vertices.
• Initialize the output sequence of vertices to be empty.
• While Σ is non-empty:
  • Find and remove a vertex v from the first set in Σ
  • If the first set in Σ is now empty, remove it from Σ
  • Add v to the end of the output sequence.
  • For each edge v-w such that w still belongs to a set S in Σ:
    • If the set S containing w has not yet been replaced while processing v, create a new empty replacement set T and place it prior to S in the sequence; otherwise, let T be the set prior to S.
    • Move w from S to T, and if this causes S to become empty remove S from Σ.

Each vertex is processed once, each edge is examined only when its two endpoints are processed, and (with an appropriate representation for the sets in Σ that allows items to be moved from one set to another in constant time) each iteration of the inner loop takes only constant time. Therefore, like simpler graph search algorithms such as breadth-first search and depth-first search, this algorithm takes linear time.

The algorithm is called lexicographic breadth-first search because the order it produces is an ordering that could also have been produced by a breadth-first search, and because if the ordering is used to index the rows and columns of an adjacency matrix of a graph then the algorithm sorts the rows and columns into lexicographical order.

Applications

Chordal graphs

A graph G is defined to be chordal if its vertices have a perfect elimination ordering, an ordering such that for any vertex v the neighbors that occur later in the ordering form a clique. In a chordal graph, the reverse of a lexicographic ordering is always a perfect elimination ordering. Therefore, one can test whether a graph is chordal in linear time by the following algorithm:

• Use lexicographic breadth-first search to find a lexicographic ordering of G
• For each vertex v:
  • Let w be the neighbor of v occurring prior to v, as close to v in the sequence as possible
    • (Continue to the next vertex v if there is no such w)
  • If the set of earlier neighbors of v (excluding w itself) is not a subset of the set of earlier neighbors of w, the graph is not chordal
• If the loop terminates without showing that the graph is not chordal, then it is chordal.

This application was the original motivation that led Rose, Tarjan & Lueker (1976) to develop the lexicographic breadth first search algorithm.

Graph coloring

A graph G is said to be perfectly orderable if there is a sequence of its vertices with the property that, for any induced subgraph of G, a greedy coloring algorithm that colors the vertices in the induced sequence ordering is guaranteed to produce an optimal coloring.

For a chordal graph, a perfect elimination ordering is a perfect ordering: the number of the color used for any vertex is the size of the clique formed by it and its earlier neighbors, so the maximum number of colors used is equal to the size of the largest clique in the graph, and no coloring can use fewer colors. An induced subgraph of a chordal graph is chordal and the induced subsequence of its perfect elimination ordering is a perfect elimination ordering on the subgraph, so chordal graphs are perfectly orderable, and lexicographic breadth-first search can be used to optimally color them.

The same property is true for a larger class of graphs, the distance-hereditary graphs: distance-hereditary graphs are perfectly orderable, with a perfect ordering given by the reverse of a lexicographic ordering, so lexicographic breadth-first search can be used in conjunction with greedy coloring algorithms to color them optimally in linear time.

Other applications

Bretscher et al. (2008) describe an extension of lexicographic breadth-first search that breaks any additional ties using the complement graph of the input graph. As they show, this can be used to recognize cographs in linear time. Habib et al. (2000) describe additional applications of lexicographic breadth-first search including the recognition of comparability graphs and interval graphs.

LexBFS ordering

An enumeration of the vertices of a graph is said to be a LexBFS ordering if it is a possible output of the application of LexBFS to this graph.

Let ${\displaystyle G=(V,E)}$ be a graph with ${\displaystyle n}$ vertices. Recall that ${\displaystyle N(v)}$ is the set of neighbors of ${\displaystyle v}$. Let ${\displaystyle \sigma =(v_1,\dots ,v_n)}$ be an enumeration of the vertices of ${\displaystyle V}$. The enumeration ${\displaystyle \sigma }$ is a LexBFS ordering (with source ${\displaystyle v_1}$) if, for all ${\displaystyle 1\le i<j<k\le n}$ with ${\displaystyle v_i\in N(v_k)\setminus N(v_j)}$, there exists ${\displaystyle m<i}$ such that ${\displaystyle v_m\in N(v_j)\setminus N(v_k)}$.