Rare Siblings Speed-Up Deterministic Detection and Counting of Small Pattern Graphs

Abstract

We consider a class of pattern graphs on $q\ge 4$ vertices that have $q-2$ distinguished vertices with equal neighborhood in the remaining two vertices. Two pattern graphs in this class are siblings if they differ by some edges connecting the distinguished vertices.

In particular, we show that if induced copies of siblings to a pattern graph in such a class are rare in the host graph then one can detect the pattern graph relatively efficiently. For example, we infer that if there are $N_d$ induced copies of a diamond (i.e., a graph on four vertices missing a single edge to be complete) in the host graph, then an induced copy of the complete graph on four vertices, $K_4,$ as well as an induced copy of the cycle on four vertices, $C_4,$ can be deterministically detected in $O(n^{2.75}+N_d)$ time. Note that the fastest known algorithm for $K_4$ and the fastest known deterministic algorithm for $C_4$ run in $O(n^{3.257})$ time. We also show that if there is a family of siblings whose induced copies in the host graph are rare then there are good chances to determine the numbers of occurrences of induced copies for all pattern graphs on q vertices relatively efficiently.

Appendix: Proof of Lemma 1

1.1 Notation

A set of single representatives of all isomorphism classes for graphs on k vertices is denoted by $\mathcal {H}_k$ while its subset consisting of graphs having an independent set on at least $k-l\ge 1$ vertices is denoted by $\mathcal {H}_k(l).$

Let H be a graph on k vertices and let $H_{sub}$ be an induced subgraph of H on l vertices such that the $k-l$ vertices in $H\setminus H_{sub}$ form an independent set. Consider the family of all supergraphs $H'$ of H (including H) in $\mathcal {H}_k$ such that $H'$ has the same vertex set as H, $H_{sub}$ is also an induced subgraph of $H'$, and the set of edges between $H_{sub}$ and $H'\setminus H_{sub}$ is the same as that between $H_{sub}$ and $H\setminus H_{sub}.$ This family is denoted by $\mathcal {H}_k(H_{sub},H),$ and its intersection with $\mathcal {H}_k$ is denoted by $\mathcal {SH}_k(H_{sub},H)$.

For a graph $H\in \mathcal {H}_k$ and a host graph G on at least k vertices, the number of sets of k vertices in G that induce a subgraph of G isomorphic to H is denoted by NI(H, G).

For $H\in \mathcal {H}_k(l),$ the set Eq(H, l) consists of the following equations in one-to-one correspondence with induced subgraphs $H_{sub}$ of H on l vertices

$$\sum _{H'\in \mathcal {SH}_k(H_{sub},H)}B(H_{sub},H')x_{H',G}= \sum _{H'\in \mathcal {SH}_k(H_{sub},H)}B(H_{sub},H')NI(H',G),$$

where $H\setminus H_{sub}$ is an independent set in H,  and $B(H_{sub},H')$ are easily computable coefficients. For our purposes, we need to define the coefficients only when $H\in \mathcal {H}_k(k-2),$ $H\setminus H_{sub}$ consists of two independent vertices and $H'\in \mathcal {SH}_k(H_{sub},H)$. Then, the coefficient $B(H_{sub},H')$ is just the number of automorphisms of $H'$ divided by the number of automorphisms of $H'$ that are identity on $H_{sub}$ by Lemma 3.6 in [11]. By Lemma 3.5 in [11], for $H\in \mathcal {H}_k(l),$ the right-hand side of an equation in Eq(H, l) can be evaluated in time $O(n^{l}(k-l)+T_l(n))$, where $T_l(n)$ stands for the time required to solve the so called l-neighborhood problem. By Theorem 6.1 in [11], $T_l(n)=O(n^{\omega (\lceil (k-l)/2 \rceil ,1,\lfloor (k-l)/2 \rfloor })$.

By SEq(G, k, l), we shall denote the system of equations obtained by picking, for each H in $\mathcal {H}_k(l),$ an arbitrary equation from Eq(H, l). By Lemma 3.7 in [11], the resulting system of $|\mathcal {H}_k(l)|$ equations is linearly independent.

1.2 5.2 Proof

Consider Theorem 4.1 in [11] with fixed $k=q$, $l=q-2$, and $O(n^{\omega (\lceil (q-2)/2 \rceil ,1,\lfloor (q-2)/2 \rfloor })$ substituted for $T_{q-2}(n)$ according to Theorem 6.1 in [11]. Then, the theorem states that if for all $H\in \mathcal {H}_{q-2}\setminus \mathcal {H}_{q-2}(q-2)$ the values NI(H, G) are known then for all $H'\in \mathcal {H}_{q-2},$ the numbers $NI(H',G)$ and $N(H',G)$ can be determined in time $O(n^{\omega (\lceil (q-2)/2 \rceil ,1,\lfloor (q-2)/2 \rfloor })$. Since $H_q\setminus H_q(q-2)=\{K_q\}$, it follows directly from Theorem 4.1 in [11] that if the number of (induced) subgraphs isomorphic to $K_q$ in the host graph is known then for all the pattern graphs on q vertices the corresponding numbers can be computed in the time specified in the lemma statement. The argumentation given in the proof of Theorem 4.1 in [11] works equally well when the number of induced copies of an arbitrary pattern graph H on q vertices is known.

Namely, following the proof of Theorem 4.1 in [11] with $k=q$ and $l=q-2,$ we form $SEq(G,q,q-2).$ Since we assume that q is fixed, the coefficients $B(H_{sub},H')$ on the left-sides of the equations in $SEq(G,q,q-2)$ can be computed in O(1) time. By Lemma 3.5 and Theorem 6.1 in [11], the right-sides of the equations can be computed in time $O(n^{\omega (\lceil (q-2)/2 \rceil ,1,\lfloor (q-2)/2 \rfloor }) .$ Let us the graphs in $\mathcal {H}_k$ so that the number of edges is non-decreasing and the graphs in $\mathcal {H}_k(l)$ form a prefix of the sorted sequence. Let B be the $|\mathcal {H}_k(l)|\times |\mathcal {H}_k|$ matrix corresponding to the left-hand sides of the equations in $SEq(G,q,q-2)$, with the rows of B corresponding to $H\in \mathcal {H}_k(q-2)$ and the columns of B corresponding to $H'\in \mathcal {H}_{q}$ sorted in the aforementioned way. Consider the leftmost maximal square submatrix M of the matrix B. Since M has zeros below the diagonal starting from the leftmost top-left corner, we infer that the resulting $|H_q(q-2)|$ equations with $|H_q|$ unknowns are also linearly independent. Hence, when we substitute the known number of induced copies of H for the variable $x_{H,G}$ corresponding to H in the equations, we obtain a system S of $|H_q(q-2)|$ equations with $|H_q|-1$ unknowns. Since $H_q\setminus H_q(q-2)=\{K_q\}$, the number of unknowns is equal to the number of equations. It follows from Remark 1 that the system S of equations resulting from the substitution is independent. Also, none of the resulting equations can disappear after the substitution. Hence, we can solve them in $O(|H_q(q-2)|^3)=O(1)$ time.    $\square$