Minimum Reload Cost Graph Factors

The concept of Reload cost in a graph refers to the cost that occurs while traversing a vertex via two of its incident edges. This cost is uniquely determined by the colors of the two edges. This concept has various applications in transportation networks, communication networks, and energy distribution networks. Various problems using this model are defined and studied in the literature. The problem of finding a spanning tree whose diameter with respect to the reload costs is the smallest possible, the problems of finding a path, trail or walk with minimum total reload cost between two given vertices, problems about finding a proper edge coloring of a graph such that the total reload cost is minimized, the problem of finding a spanning tree such that the sum of the reload costs of all paths between all pairs of vertices is minimized, and the problem of finding a set of cycles of minimum reload cost, that cover all the vertices of a graph, are examples of such problems. % In this work we focus on the last problem. Noting that a cycle cover of a graph is a 2-factor of it, we generalize the problem to that of finding an $r$-factor of minimum reload cost of an edge colored graph. We prove several NP-hardness results for special cases of the problem. Namely, bounded degree graphs, planar graphs, bounded total cost, and bounded number of distinct costs. For the special case of $r=2$, our results imply an improved NP-hardness result. On the positive side, we present a polynomial-time solvable special case which provides a tight boundary between the polynomial and hard cases in terms of $r$ and the maximum degree of the graph. We then investigate the parameterized complexity of the problem, prove W[1]-hardness results and present an FPT algorithm

Cohomology computations for Artin groups, Bestvina-Brady groups, and graph products

We compute: * the cohomology with group ring coefficients of Artin groups (or actually, of their associated Salvetti complexes), Bestvina-Brady groups, and graph products of groups, * the L^2-Betti numbers of Bestvina-Brady groups and of graph products of groups, * the weighted L^2-Betti numbers of graph products of Coxeter groups. In the case of arbitrary graph products there is an additional proviso: either all factors are infinite or all are finite.(However, for graph products of Coxeter groups this proviso is unnecessary.)Comment: 56 page

The Tutte's condition in terms of graph factors

Let $G$ be a connected general graph of even order, with a function $f\colon V(G)\to\Z^+$. We obtain that $G$ satisfies the Tutte's condition $$o(G-S)\le \sum_{v\in S}f(v)\qquad\text{for any nonempty set $S\subset V(G)$},$$ with respect to $f$ if and only if $G$ contains an $H$-factor for any function $H\colon V(G)\to 2^\N$ such that $H(v)\in \{J_f(v),\,J_f^+(v)\}$ for each $v\in V(G)$, where the set $J_f(v)$ consists of the integer $f(v)$ and all positive odd integers less than $f(v)$, and the set $J^+_f(v)$ consists of positive odd integers less than or equal to $f(v)+1$. We also obtain a characterization for graphs of odd order satisfying the Tutte's condition with respect to a function.Comment: 5 page

Abelian 1-factorizations of complete multipartite graphs

An automorphism group G of a 1-factorization of the complete multipartite graph $K_{m\times n}$ consists in permutations of the vertices of the graph mapping factors to factors. In this paper, we give a complete answer to the existence or non-existence problem of a 1-factorization of $K_{m\times n}$ admitting an abelian group acting sharply transitively on the vertices of the graph.Comment: 7 page

The Relaxed Square Property

Graph products are characterized by the existence of non-trivial equivalence relations on the edge set of a graph that satisfy a so-called square property. We investigate here a generalization, termed RSP-relations. The class of graphs with non-trivial RSP-relations in particular includes graph bundles. Furthermore, RSP-relations are intimately related with covering graph constructions. For K_23-free graphs finest RSP-relations can be computed in polynomial-time. In general, however, they are not unique and their number may even grow exponentially. They behave well for graph products, however, in sense that a finest RSP-relations can be obtained easily from finest RSP-relations on the prime factors

Local algorithms for the prime factorization of strong product graphs

The practical application of graph prime factorization algorithms is limited in practice by unavoidable noise in the data. A first step towards error-tolerant "approximate" prime factorization, is the development of local approaches that cover the graph by factorizable patches and then use this information to derive global factors. We present here a local, quasi-linear al- gorithm for the prime factorization of "locally unrefined" graphs with respect to the strong product. To this end we introduce the backbone B(G) for a given graph G and show that the neighborhoods of the backbone vertices provide enough information to determine the global prime factors

Unique factorisation of additive induced-hereditary properties

An additive hereditary graph property is a set of graphs, closed under isomorphism and under taking subgraphs and disjoint unions. Let ${\cal P}_1, >..., {\cal P}_n$ be additive hereditary graph properties. A graph $G$ has property $({\cal P}_1 \circ ... \circ {\cal P}_n)$ if there is a partition $(V_1, ..., V_n)$ of $V(G)$ into $n$ sets such that, for all $i$, the induced subgraph $G[V_i]$ is in ${\cal P}_i$. A property ${\cal P}$ is reducible if there are properties ${\cal Q}$, ${\cal R}$ such that ${\cal P} = {\cal Q} \circ {\cal R}$; otherwise it is irreducible. Mih\'{o}k, Semani\v{s}in and Vasky [J. Graph Theory {\bf 33} (2000), 44--53] gave a factorisation for any additive hereditary property ${\cal P}$ into a given number $dc({\cal P})$ of irreducible additive hereditary factors. Mih\'{o}k [Discuss. Math. Graph Theory {\bf 20} (2000), 143--153] gave a similar factorisation for properties that are additive and induced-hereditary (closed under taking induced-subgraphs and disjoint unions). Their results left open the possiblity of different factorisations, maybe even with a different number of factors; we prove here that the given factorisations are, in fact, unique.Comment: 26 pages, 4 figures, to appear in Discussiones Mathematicae Graph Theor

Distance-residual graphs

If we are given a connected finite graph $G$ and a subset of its vertices $V_{0}$, we define a distance-residual graph as a graph induced on the set of vertices that have the maximal distance from $V_{0}$. Some properties and examples of distance-residual graphs of vertex-transitive, edge-transitive, bipartite and semisymmetric graphs are shown. The relations between the distance-residual graphs of product graphs and their factors are shown.Comment: 14 pages, 4 figure

Triangulating planar graphs while keeping the pathwidth small

Any simple planar graph can be triangulated, i.e., we can add edges to it, without adding multi-edges, such that the result is planar and all faces are triangles. In this paper, we study the problem of triangulating a planar graph without increasing the pathwidth by much. We show that if a planar graph has pathwidth $k$, then we can triangulate it so that the resulting graph has pathwidth $O(k)$ (where the factors are 1, 8 and 16 for 3-connected, 2-connected and arbitrary graphs). With similar techniques, we also show that any outer-planar graph of pathwidth $k$ can be turned into a maximal outer-planar graph of pathwidth at most $4k+4$. The previously best known result here was $16k+15$.Comment: To appear (without the appendix) at WG 201

Some Criteria for a Signed Graph to Have Full Rank

A weighted graph $G^{\omega}$ consists of a simple graph $G$ with a weight $\omega$, which is a mapping,$\omega$: $E(G)\rightarrow\mathbb{Z}\backslash\{0\}$. A signed graph is a graph whose edges are labeled with $-1$ or $1$. In this paper, we characterize graphs which have a sign such that their signed adjacency matrix has full rank, and graphs which have a weight such that their weighted adjacency matrix does not have full rank. We show that for any arbitrary simple graph $G$, there is a sign $\sigma$ so that $G^{\sigma}$ has full rank if and only if $G$ has a $\{1,2\}$-factor. We also show that for a graph $G$, there is a weight $\omega$ so that $G^{\omega}$ does not have full rank if and only if $G$ has at least two $\{1,2\}$-factors