Induced Subforests and Superforests

TL;DR

It is shown that finding a maximum subforest is NP-hard already for two subdivided stars while finding a minimum superforest is tractable for two trees but NP-hard for three trees.

Abstract

Graph isomorphism, subgraph isomorphism, and maximum common subgraphs are classical well-investigated objects. Their (parameterized) complexity and efficiently tractable cases have been studied. In the present paper, for a given set of forests, we study maximum common induced subforests and minimum common induced superforests. We show that finding a maximum subforest is NP-hard already for two subdivided stars while finding a minimum superforest is tractable for two trees but NP-hard for three trees. For a given set of $k$ trees, we present an efficient greedy $\left(\frac{k}{2}-\frac{1}{2}+\frac{1}{k}\right)$-approximation algorithm for the minimum superforest problem. Finally, we present a polynomial time approximation scheme for the maximum subforest problem for any given set of forests.

Figures (7)

• Figure 1: The tree $T(x_i)$ if $q=4$ and $x_i$ is contained in the three triples $(x_i,y_2,z_1)$, $(x_i,y_3,z_4)$, and $(x_i,y_4,z_2)$.
• Figure 2: The left shows $T(x_i)$ if $q=4$ and $x_i$ is contained in exactly the two triples $(x_i,y_3,z_4)$ and $(x_i,y_4,z_2)$. The right shows $T(x_i)$ if $q=4$ and $x_i$ is contained in only one triple $(x_i,y_4,z_2)$.
• Figure 3: $T(y_3)$ on the left and $T(z_2)$ on the right.
• Figure 4: Three trees $T_1$, $T_2$, and $T_3$.
• Figure 5: A supertree for $\{ T_1,T_2,T_3\}$.

Theorems & Definitions (13)

• Proposition 1
• Proposition 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• proof : Proof of Proposition \ref{['proposition1']}
• proof : Proof of Theorem \ref{['theorem1']}
• proof : Proof of Theorem \ref{['theorem5']}
• proof : Proof of Theorem \ref{['theorem2']}