{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:19Z","timestamp":1753893859618,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $G=(V,E)$ be a simple undirected graph with $n$ vertices then a set partition $\\pi=\\{V_1, \\ldots, V_k\\}$ of the vertex set of $G$ is a connected set partition if each subgraph $G[V_j]$ induced by the blocks $V_j$ of $\\pi$ is connected for $1\\le j\\le k$. Define $q_{i}(G)$ as the number of connected set partitions in $G$ with $i$ blocks. The partition polynomial is $Q(G, x)=\\sum_{i=0}^n q_{i}(G)x^i$. This paper presents a splitting approach to the partition polynomial on a separating vertex set $X$ in $G$ and summarizes some properties of the bond lattice. Furthermore the bivariate partition polynomial $Q(G,x,y)=\\sum_{i=1}^n \\sum_{j=1}^m q_{ij}(G)x^iy^j$ is briefly discussed, where $q_{ij}(G)$ counts the number of connected set partitions with $i$ blocks and $j$ intra block edges. Finally the complexity for the bivariate partition polynomial is proven to be $\\sharp P$-hard.<\/jats:p>","DOI":"10.37236\/501","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:57:47Z","timestamp":1578715067000},"source":"Crossref","is-referenced-by-count":0,"title":["Counting Connected Set Partitions of Graphs"],"prefix":"10.37236","volume":"18","author":[{"given":"Frank","family":"Simon","sequence":"first","affiliation":[]},{"given":"Peter","family":"Tittmann","sequence":"additional","affiliation":[]},{"given":"Martin","family":"Trinks","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2010,1,12]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p14\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p14\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T00:02:16Z","timestamp":1579305736000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p14"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1,12]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/501","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2010,1,12]]},"article-number":"P14"}}