Finding Biclique Partitions of Co-Chordal Graphs

The biclique partition number $(\operatorname{bp})$ of a graph $G$ is referred to as the least number of complete bipartite (biclique) subgraphs that are required to cover the edges of the graph exactly once (however, the vertices can belong to two or more bicliques). The set of such biclique subgraphs is called a biclique partition of $G$. Graham and Pollak first introduced this concept in network addressing [11] and graph storage [12]. Their famous Graham-Pollak Theorem proves a result about the biclique partition numbers on complete graphs and it draws much attention from algebraic graph theory [7, 17, 19, 25, 26]. However, no purely combinatorial proof is known to the result [18]. Rawshdeh and Al-Ezeh [21] extended Graham-Pollak Theorem to find biclique partition numbers on line graphs and their complements of complete graphs and bicliques. The biclique partition also has a strong connection with biclique cover number $(\operatorname{bc})$, where the edges of a graph are covered by bicliques but not necessarily disjointed. Pinto [20] showed that $\operatorname{bp}(G)\leq\frac{1}{2}(3^{\operatorname{bc}(G)}-1)$.

Moreover, finding a biclique partition with the minimum size is NP-complete even on the graphs without 4-cycles [15]. In this work, we focus on studying the biclique partition number on co-chordal (complement of chordal) and its important subclass, split graphs¹¹1Almost all chordal graphs are split graphs. It means that as $n$ goes to infinity, the fraction of $n$-vertex split graphs in $n$-vertex chordal graphs goes to 1 [4]. Since the complement of a split graph is also split, a split graph is also co-chordal..

There are also many related research studies around biclique partitions. Motivated by a technique for clustering data on binary matrices, Bein et al. [3] considered a biclique vertex partition problem on a bipartite graph where each vertex is covered exactly once in a collection of biclique subgraphs. De Sousa Filho et al. [9] also studied the biclique vertex partition problem and its variant bicluster editing problem, where they developed a polyhedral study on biclique vertex partitions on a complete bipartite graph. Groshaus et al. [13] gave a polynomial-time algorithm to determine whether a graph is a biclique graph (an intersection graph of the bicliques) of a subclass of split graphs. Shigeta and Amano [24] provided an explicit construction of an ordered biclique partition, a variant of biclique partition, of $K_{n}$ of size $n^{1/2+o(1)}$, which improved the $O(n^{2/3})$ bound shown by Amano [2].

Another related graph characteristic is $\operatorname{bp}_{k}(G)$ where the edges can be covered by at least one and at most $k$ biclique subgraphs [1] and Alon showed that the minimum possible number for $\operatorname{bp}_{k}(K_{n})$ is $\Theta(kn^{1/k})$, where $K_{n}$ is a complete graph with $n$ vertices. Recent work by Rohatgi et al. [22] showed that if each edge is exactly covered by $k$ bicliques, the number of bicliques required to cover $K_{n}$ is $(1+o(1))n$.

In this paper, we study a biclique (edge) partition problem on co-chordal graphs and split graphs using clique trees, moplexes, and lexicographic breadth-first search (LexBFS) defined in Section 2. We introduce a new definition, partitioning biclique, which can naturally naturally partition a graph into two induced subgraphs with no shared edges in Section 3. In Section 4, we provide a heuristic to find a biclique partition on a co-chordal graph given a clique tree of the complement of the co-chordal graph. We also prove the correctness of the heuristic and show the size of the biclique partition is exactly equal to the number of maximal cliques in the complementary graph of the co-chordal graph minus one. We also provide a corollary that states that the biclique partition number of a co-chordal graph is less than the number of maximal cliques of its complement. In Section 5, we provide an efficient heuristic to obtain biclique partitions of co-chordal graphs by finding moplexes [5], defined later, using LexBFS. We also show that two heuristics provide biclique partitions of the same size. In Section 6, we prove that both heuristics can find a minimum biclique partition of $G$ if its complement $G^{c}$ is chordal and clique vertex irreducible. We also derive a lower bound of the biclique partition number of split graphs and show that our heuristics can obtain a biclique partition on any split graph with a size no more than the biclique partition number plus one. In Section 7, we summarize the contribution of our work and point out some future directions.

A simple graph is a pair $G=(V,E)$ where $V$ is a finite set of vertices and $E\subseteq\{uv:u,v\in V,u\neq v\}$. We use $V(G)$ and $E(G)$ to represent the vertex set and edge set of the graph $G$. Two vertices are adjacent in $G$ if there is an edge between them. A subgraph $G^{\prime}=(V^{\prime},E^{\prime})$ of $G$ is a graph where $V^{\prime}\subseteq V$ and $E^{\prime}\subseteq\{uv\in E:u,v\in V^{\prime}\}$. Given $A\subseteq V$, the subgraph of $G$ induced by $A$ is denoted as $G(A)=(A,E_{A})$, where $E_{A}=\{uv\in E:u,v\in A\}$. A clique is a subset of vertices of an graph $G$ such that every two distinct vertices are adjacent in $G$. The neighborhood of a vertex $v$ of a graph $G$ is the set of all vertices, other than $v$, that are adjacent with $v$ and is denoted as $N_{G}(v)$. The closed neighborhood of $v$ is denoted as $N_{G}[v]=N_{G}(v)\cup\{v\}$. For simplicity, we also define the neighborhood of a vertex set $A$ of a graph $G$ as $N_{G}(A)=\{u\in N_{G}(v):\forall v\in A\}\setminus A$ and closed neighborhood of $A$ as $N_{G}[A]=N_{G}(A)\cup A$. A maximal clique of $G$ is a clique of $G$ such that it is not a proper subset of any clique of $G$. Note that a vertex set with only one vertex $K_{1}$ is also a clique. We denote the number of maximal cliques of $G$ as $\operatorname{mc}(G)$ and we use $\mathcal{K}_{G}$ or $\mathcal{K}$ to denote the set of all maximal cliques of $G$. A maximum clique of $G$ is a clique of $G$ with the maximum number of vertices and we denote that number as the clique number of $G$, $\omega(G)$. An independent set of a graph $G$ is a set of vertices that are pairwise nonadjacent with each other in $G$. Similarly, a maximal independent set is an independent set that is not a proper subset of any independent set and a maximum independent set is an independent set with the maximum number of vertices. Note that an independent set can be empty or only have one vertex.

A graph $C_{n}=(V,E)$ is a cycle if the vertices and edges: $V=\{v_{1},v_{2},\ldots,v_{n}\}$ and $E=\{v_{1}v_{2},v_{2}v_{3},\ldots,v_{n-1}v_{n},v_{n}v_{1}\}$. A graph is a tree if it is connected and does not have any subgraph that is a cycle.

Given two vertex sets $U$ and $V$, we denote $U\times V$ to be the edge set $\{uv:u\in U,v\in V\}$. A bipartite graph $G=(L\cup R,E)$ is a graph where $L$ and $R$ are disjointed vertex sets with the edge set $E\subseteq L\times R$. A biclique graph is a complete bipartite graph $G=(L\cup R,E)$ where $E=L\times R$ and we denote it as $\{L,R\}$ for short. A biclique partition of a graph $G$ is a collection of biclique subgraphs of $G$ such that every edge of $G$ is in exactly one biclique of the collection. The minimum biclique partition problem on $G$ is to find a biclique partition with the minimum number of bicliques in the collection and we denote that value to be $\operatorname{bp}(G)$.

We denote that $\llbracket n\rrbracket=\{1,2,\ldots,n\}$ where $n$ is a positive integer. A vertex is simplicial if its neighborhood is a clique. An ordering $v_{1},v_{2},\ldots,v_{n}$ of $V$ is a perfect elimination ordering if for all $i\in\llbracket n\rrbracket$, $v_{i}$ is simplicial on the induced subgraph $G(\{v_{j}:j\in\{i,i+1,\ldots,n\}\})$. An ordering function $\sigma:\llbracket n\rrbracket\rightarrow V$ is defined to describe the ordering of vertices $V$. A clique tree $\mathcal{T}_{\mathcal{K}}$ for a chordal graph $G$ is a tree where each vertex represents a maximal clique of $G$ and satisfies the clique-intersection property: given any two distinct maximal cliques $K_{1}$ and $K_{2}$ in the tree, every clique on the path between $K_{1}$ and $K_{2}$ in $\mathcal{T}_{\mathcal{K}}$ contains $K_{1}\cap K_{2}$. We also define the middle set of edge $e\in\mathcal{T}_{\mathcal{K}}$, $\operatorname{mid}(e)$, to be the intersection of the vertices of cliques on its two ends.

A graph $G$ is clique vertex irreducible if every maximal clique in $G$ has a vertex which does not lie in any other maximal clique of $G$ [16].

A split graph is a graph whose vertices can be partitioned into a clique and an independent set. A graph is chordal if there is no induced cycle subgraph of length greater than 3. Note that a split graph is also chordal. A graph is chordal if and only if it has a clique tree [6]. Also, a graph is chordal if and only if it has a perfect elimination ordering [10]. A perfect elimination ordering of a chordal graph can be obtained by lexicographic breadth-first search (LexBFS) described in Algorithm 1 [8, 23], where lexicographical order is defined in Definition 1. Note that the LexBFS algorithm in Algorithm 1 can be implemented in linear-time: $O(|V|+|E|)$ with partition refinement [14].

In our work, we study biclique partitions on co-chordal graphs, so most of the $G^{c}$’s in the following sections are chordal graphs. We use $\mathcal{K}^{c}$ and $\mathcal{T}_{\mathcal{K}^{c}}$ to denote the set of all maximal cliques and a clique tree of $G^{c}$.

A module of a graph $G=(V,E)$ is a vertex set $A\subseteq V$ such that all the vertices in $A$ share the same neighborhood in $V\setminus A$. A separator of $G=(V,E)$ is a set of vertices, say $S$, such that $G(V\setminus S)$ is disconnected. A separator $S$ is minimal if no proper set of $S$ is a separator. A moplex $X$ of $G$ is both a clique and a module such that $N_{G}(X)$ is a minimal separator (see Figure 1). Berry and Bordat [5] discovered that LexBFS can be applied to find a moplex of a general graph $G$ in linear time, which motivates us to propose the biclique partition algorithm in Section 5.

We will introduce a new definition: partitioning biclique, which can naturally partition the edges of a graph into two edge disjoint subgraphs. We first start with a definition of partitioning biclique.

It is worth to noticing that $\{L,R\}$ is not necessarily equal to the induced subgraph $G(L\cup R)$. It means that $L$ and $R$ might not be independent sets of $G$.

We then show that an arbitrary biclique subgraph can divide the edges of the original graphs into three parts: the biclique and two induced subgraphs. See Figure 2 for a visualization of this idea: the edges of graph $G$ are partitioned into the edges of $\{L,R\}$, $G(V\setminus L)$, and $G(V\setminus R)$.

Next, we show that a biclique subgraph is a partitioning biclique of a graph $G$ if and only if the vertices in $G$ that are not in the biclique form an independent set in $G$.

Note that an independent set can be empty. Since the partitioning biclique can partition the original graph into a biclique subgraph and two induced subgraphs, it can be used to design a heuristic to find a biclique partition of a graph. In the next two sections, we will focus on a class of graph, co-chordal, where partitioning bicliques are easy to find since the complementary graph is chordal.

In this section, we want to design a heuristic to find a biclique partition of a co-chordal graph $G$. Since $G^{c}$ is chordal, one of the good ways to represent $G^{c}$ is its clique tree where each vertex represents a maximal clique of $G$ and satisfies the clique-intersection property. We demonstrate a heuristic with an input of a clique tree $\mathcal{T}_{\mathcal{K}^{c}}$ of $G^{c}$ and an output of a biclique partition of $G$ in Algorithm 2. We also show that the size of that biclique partition is equal to $\operatorname{mc}(G^{c})-1$, which provides us an upper bound of the biclique partition number of co-chordal graphs.

We prove that the output of Algorithm 2 is a biclique partition of a co-chordal graph $G$ by showing that at each recursion a nonempty partitioning biclique $\{L,R\}$ is found and two subtrees $\mathcal{T}_{\mathcal{K}^{c}_{1}}$ and $\mathcal{T}_{\mathcal{K}^{c}_{2}}$ are also clique trees of two induced subgraphs $G^{c}(V\setminus L)$, and $G^{c}(V\setminus R)$ of $G^{c}$ respectively. Note that the edge $e$ can be selected arbitrarily in Algorithm 2.

We next prove that $\{L,R\}$ is a partitioning biclique of $G$. Note that $L$ and $R$ do not have to be independent sets.

In Claim 2, we prove part (2) of Proposition 3. We first remark that any subgraph of a chordal graph is chordal and then use it to prove part (2) in Claim 2.

Next, we show that the edge set of biclique $\{L,R\}$ is not empty.

Next, we show that the output of Algorithm 2 is a biclique partition of a co-chordal graph $G$.

Finally, we prove that the size of the output of Algorithm 2 is equal to one less than the number of maximal cliques of $G^{c}$.

Theorem 2 leads us to a direct result that the biclique partition number of a co-chordal graph $G$ is less than the number of maximal cliques of $G^{c}$.

The total time complexity of Algorithm 2 is $O[|V|^{3}]$, which includes both of taking the complement of a co-chordal graph $G$ and computing a clique tree of $G^{c}$.

Although Algorithm 2 can give a biclique partition of a co-chordal graph $G$, it requires a clique tree of its complementary chordal graph $G^{c}$ and computes unions of some maximal cliques, which could include some redundant computations. Instead of using clique tree, we can use a moplex in $G^{c}$ to find a biclique partition. Figure 3 motivates the idea via a simple example. In Algorithm 2, we can select an arbitrary edge in the clique tree to find a biclique. Thus, we can always select an edge that is incident to a leaf node of the tree. For example, if we select the edge with middle set $\{x_{3}\}$, the biclique built in the current call of FindPartition($\mathcal{T}$) has $L=\{x_{4},x_{5},x_{6},x_{7}\}$ and $R=\{x_{1},x_{2}\}$, where $\{x_{1},x_{2}\}$ is a moplex of $G^{c}$. We can also see that selecting the other two edges can also lead to bicliques with a moplex as $L$ or $R$. It motivates us to design another heuristic to find biclique partitions on co-chordal graphs by finding moplexes, which can be found efficiently by LexBFS. In this section, we show that Algorithm 3 can return a biclique partition of a co-chordal graph $G$ with a same size as Algorithm 2 in $O[|V|(|V|+|E^{c}|)]$ time.

Note that Xu et al.[27] also apply lexicographical depth-first search (LexDFS) to find moplexes, which can be an alternative method to find moplexes than LexBFS in Algorithm 3.

Before we can prove that Algorithm 3 returns a biclique partition of a co-chordal graph, we need to show that $\{L,R\}$ is a partitioning biclique of $G^{\prime c}$ with vertices $V^{\prime}$. In addition, one of the two induced subgraphs, $G^{\prime c}(V^{\prime}\setminus L)$, has an empty edge set.

Next, we show that the output of Algorithm 3 is a biclique partition of a co-chordal graph $G$ with a size of $\operatorname{mc}(G^{c})-1$.

We also show that Algorithm 3 is efficient: $O[|V|(|V|+|E^{c}|)]$ time.