4,961 research outputs found

    Graph homomorphisms between trees

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    In this paper we study several problems concerning the number of homomorphisms of trees. We give an algorithm for the number of homomorphisms from a tree to any graph by the Transfer-matrix method. By using this algorithm and some transformations on trees, we study various extremal problems about the number of homomorphisms of trees. These applications include a far reaching generalization of Bollob\'as and Tyomkyn's result concerning the number of walks in trees. Some other highlights of the paper are the following. Denote by hom(H,G)\hom(H,G) the number of homomorphisms from a graph HH to a graph GG. For any tree TmT_m on mm vertices we give a general lower bound for hom(Tm,G)\hom(T_m,G) by certain entropies of Markov chains defined on the graph GG. As a particular case, we show that for any graph GG, exp(Hλ(G))λm1hom(Tm,G),\exp(H_{\lambda}(G))\lambda^{m-1}\leq\hom(T_m,G), where λ\lambda is the largest eigenvalue of the adjacency matrix of GG and Hλ(G)H_{\lambda}(G) is a certain constant depending only on GG which we call the spectral entropy of GG. In the particular case when GG is the path PnP_n on nn vertices, we prove that hom(Pm,Pn)hom(Tm,Pn)hom(Sm,Pn),\hom(P_m,P_n)\leq \hom(T_m,P_n)\leq \hom(S_m,P_n), where TmT_m is any tree on mm vertices, and PmP_m and SmS_m denote the path and star on mm vertices, respectively. We also show that if TmT_m is any fixed tree and hom(Tm,Pn)>hom(Tm,Tn),\hom(T_m,P_n)>\hom(T_m,T_n), for some tree TnT_n on nn vertices, then TnT_n must be the tree obtained from a path Pn1P_{n-1} by attaching a pendant vertex to the second vertex of Pn1P_{n-1}. All the results together enable us to show that |\End(P_m)|\leq|\End(T_m)|\leq|\End(S_m)|, where \End(T_m) is the set of all endomorphisms of TmT_m (homomorphisms from TmT_m to itself).Comment: 47 pages, 15 figure

    The Complexity of Counting Homomorphisms to Cactus Graphs Modulo 2

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    A homomorphism from a graph G to a graph H is a function from V(G) to V(H) that preserves edges. Many combinatorial structures that arise in mathematics and computer science can be represented naturally as graph homomorphisms and as weighted sums of graph homomorphisms. In this paper, we study the complexity of counting homomorphisms modulo 2. The complexity of modular counting was introduced by Papadimitriou and Zachos and it has been pioneered by Valiant who famously introduced a problem for which counting modulo 7 is easy but counting modulo 2 is intractable. Modular counting provides a rich setting in which to study the structure of homomorphism problems. In this case, the structure of the graph H has a big influence on the complexity of the problem. Thus, our approach is graph-theoretic. We give a complete solution for the class of cactus graphs, which are connected graphs in which every edge belongs to at most one cycle. Cactus graphs arise in many applications such as the modelling of wireless sensor networks and the comparison of genomes. We show that, for some cactus graphs H, counting homomorphisms to H modulo 2 can be done in polynomial time. For every other fixed cactus graph H, the problem is complete for the complexity class parity-P which is a wide complexity class to which every problem in the polynomial hierarchy can be reduced (using randomised reductions). Determining which H lead to tractable problems can be done in polynomial time. Our result builds upon the work of Faben and Jerrum, who gave a dichotomy for the case in which H is a tree.Comment: minor change

    Preservation and decomposition theorems for bounded degree structures

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    We provide elementary algorithms for two preservation theorems for first-order sentences (FO) on the class \^ad of all finite structures of degree at most d: For each FO-sentence that is preserved under extensions (homomorphisms) on \^ad, a \^ad-equivalent existential (existential-positive) FO-sentence can be constructed in 5-fold (4-fold) exponential time. This is complemented by lower bounds showing that a 3-fold exponential blow-up of the computed existential (existential-positive) sentence is unavoidable. Both algorithms can be extended (while maintaining the upper and lower bounds on their time complexity) to input first-order sentences with modulo m counting quantifiers (FO+MODm). Furthermore, we show that for an input FO-formula, a \^ad-equivalent Feferman-Vaught decomposition can be computed in 3-fold exponential time. We also provide a matching lower bound.Comment: 42 pages and 3 figures. This is the full version of: Frederik Harwath, Lucas Heimberg, and Nicole Schweikardt. Preservation and decomposition theorems for bounded degree structures. In Joint Meeting of the 23rd EACSL Annual Conference on Computer Science Logic (CSL) and the 29th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS'14, pages 49:1-49:10. ACM, 201
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