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

Abstract

Let t be a nonnegative integer and G = (V(G),E(G)) be a graph. For v ∈ V(G), let NG(v) = {u ∈ V(G) \ {v} : uv ∈ E(G)}. And for S ⊆ V(G), we define dS(G; v) = |NG(v) ∩ S| for v ∈ S and dS(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)4n) time, where n = |V(G)|. Moreover, we give a polynomial-time algorithm for this problem on graphs with bounded treewidth.