Maximum weight t-sparse set problem on vector-weighted graphs

Let t be a nonnegative integer and G = ( V ( G ), E ( G )) be a graph. For v ∈ V ( G ), let N G ( v ) = { u ∈ V ( G ) \ { v } : uv ∈ E ( G )}. And for S ⊆ V ( G ), we define d S ( G ; v ) = | N G (v) ∩ S | for v ∈ S and d S ( G ; v ) = −1 for v ∈ V ( G ) \ S . A subset S ⊆ V ( G ) is called a t -sparse set of G if the maximum degree of the induced subgraph G [ S ] does not exceed t . In particular, a 0-sparse set is precisely an independent set. A vector-weighted graph $ (G,\vec{w},t)$ is a graph G with a vector weight function $ \vec{w}:V(G)\to {\mathbb{R}}^{t+2}$, where $ \vec{w}(v)=(w(v;-1),w(v;0),\dots,w(v;t))$ for each v ∈ V ( G ). The weight of a t -sparse set S in $ (G,\vec{w},t)$ is defined as $ \vec{w}(S,G)={\sum }_v w(v;{d}_S(G;v))$. And a t -sparse set S is a maximum weight t -sparse set of $ (G,\vec{w},t)$ if there is no t -sparse set of larger weight in $ (G,\vec{w},t)$. In this paper, we propose the maximum weight t -sparse set problem on vector-weighted graphs, which is to find a maximum weight t -sparse set of $ (G,\vec{w},t)$. We design a dynamic programming algorithm to find a maximum weight t -sparse set of an outerplane graph $ (G,\vec{w},t)$ which takes O (( t + 2) 4 n ) time, where n = | V ( G )|. Moreover, we give a polynomial-time algorithm for this problem on graphs with bounded treewidth.