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Register a new userTwo-connected graphs with prescribed three-connected components
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Added by
Pierre Leroux
Publication date
2008
fields
Informatics Engineering
and research's language is
English
Authors
Andrei Gagarin
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We adapt the classical 3-decomposition of any 2-connected graph to the case of simple graphs (no loops or multiple edges). By analogy with the block-cutpoint tree of a connected graph, we deduce from this decomposition a bicolored tree tc(g) associated with any 2-connected graph g, whose white vertices are the 3-components of g (3-connected components or polygons) and whose black vertices are bonds linking together these 3-components, arising from separating pairs of vertices of g. Two fundamental relationships on graphs and networks follow from this construction. The first one is a dissymmetry theorem which leads to the expression of the class B=B(F) of 2-connected graphs, all of whose 3-connected components belong to a given class F of 3-connected graphs, in terms of various rootings of B. The second one is a functional equation which characterizes the corresponding class R=R(F) of two-pole networks all of whose 3-connected components are in F. All the rootings of B are then expressed in terms of F and R. There follow corresponding identities for all the associated series, in particular the edge index series. Numerous enumerative consequences are discussed.
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Let $S_k(n)$ be the maximum number of orientations of an $n$-vertex graph $G$ in which no copy of $K_k$ is strongly connected. For all integers $n$, $kgeq 4$ where $ngeq 5$ or $kgeq 5$, we prove that $S_k(n) = 2^{t_{k-1}(n)}$, where $t_{k-1}(n)$ is the number of edges of the $n$-vertex $(k-1)$-partite Turan graph $T_{k-1}(n)$, and that $T_{k-1}(n)$ is the only $n$-vertex graph with this number of orientations. Furthermore, $S_4(4) = 40$ and this maximality is achieved only by $K_4$.
A signed graph is a pair $(G,Sigma)$, where $G=(V,E)$ is a graph (in which parallel edges are permitted, but loops are not) with $V={1,ldots,n}$ and $Sigmasubseteq E$. The edges in $Sigma$ are called odd and the other edges of $E$ even. By $S(G,Sigma)$ we denote the set of all symmetric $ntimes n$ matrices $A=[a_{i,j}]$ with $a_{i,j}<0$ if $i$ and $j$ are adjacent and connected by only even edges, $a_{i,j}>0$ if $i$ and $j$ are adjacent and connected by only odd edges, $a_{i,j}in mathbb{R}$ if $i$ and $j$ are connected by both even and odd edges, $a_{i,j}=0$ if $i ot=j$ and $i$ and $j$ are non-adjacent, and $a_{i,i} in mathbb{R}$ for all vertices $i$. The parameters $M(G,Sigma)$ and $xi(G,Sigma)$ of a signed graph $(G,Sigma)$ are the largest nullity of any matrix $Ain S(G,Sigma)$ and the largest nullity of any matrix $Ain S(G,Sigma)$ that has the Strong Arnold Hypothesis, respectively. In a previous paper, we gave a characterization of signed graphs $(G,Sigma)$ with $M(G,Sigma)leq 1$ and of signed graphs with $xi(G,Sigma)leq 1$. In this paper, we characterize the $2$-connected signed graphs $(G,Sigma)$ with $M(G,Sigma)leq 2$ and the $2$-connected signed graphs $(G,Sigma)$ with $xi(G,Sigma)leq 2$.
For a graph G=(V,E), the k-dominating graph of G, denoted by $D_{k}(G)$, has vertices corresponding to the dominating sets of G having cardinality at most k, where two vertices of $D_{k}(G)$ are adjacent if and only if the dominating set corresponding to one of the vertices can be obtained from the dominating set corresponding to the second vertex by the addition or deletion of a single vertex. We denote by $d_{0}(G)$ the smallest integer for which $D_{k}(G)$ is connected for all k greater than or equal to $d_{0}(G)$. It is known that $d_{0}(G)$ lies between $Gamma(G)+1$ and $|V|$ (inclusive), where ${Gamma}(G)$ is the upper domination number of G, but constructing a graph G such that $d_{0}(G)>{Gamma}(G)+1$ appears to be difficult. We present two related constructions. The first construction shows that for each integer k greater than or equal to 3 and each integer r from 1 to k-1, there exists a graph $G_{k,r}$ such that ${Gamma}(G_{k,r})=k, {gamma}(G_{k,r})=r+1$ and $d_{0}(G_{k,r})=k+r={Gamma}(G)+{gamma}(G)-1$. The second construction shows that for each integer k greater than or equal to 3 and each integer r from 1 to k-1, there exists a graph $Q_{k,r}$ such that ${Gamma}(Q_{k,r})=k, {gamma}(Q_{k,r})=r$ and $d_{0}(Q_{k,r})=k+r={Gamma}(G)+{gamma}(G)$.
A finite graph $G$ is said to be {em $(G,3)$-$($connected$)$ homogeneous} if every isomorphism between any two isomorphic (connected) subgraphs of order at most $3$ extends to an automorphism $gin G$ of the graph, where $G$ is a group of automorphisms of the graph. In 1985, Cameron and Macpherson determined all finite $(G, 3)$-homogeneous graphs. In this paper, we develop a method for characterising $(G,3)$-connected homogeneous graphs. It is shown that for a finite $(G,3)$-connected homogeneous graph $G=(V, E)$, either $G_v^{G(v)}$ is $2$--transitive or $G_v^{G(v)}$ is of rank $3$ and $G$ has girth $3$, and that the class of finite $(G,3)$-connected homogeneous graphs is closed under taking normal quotients. This leads us to study graphs where $G$ is quasiprimitive on $V$. We determine the possible quasiprimitive types for $G$ in this case and give new constructions of examples for some possible types.
Recent papers have formulated the problem of learning graphs from data as an inverse covariance estimation with graph Laplacian constraints. While such problems are convex, existing methods cannot guarantee that solutions will have specific graph topology properties (e.g., being $k$-partite), which are desirable for some applications. In fact, the problem of learning a graph with given topology properties, e.g., finding the $k$-partite graph that best matches the data, is in general non-convex. In this paper, we develop novel theoretical results that provide performance guarantees for an approach to solve these problems. Our solution decomposes this problem into two sub-problems, for which efficient solutions are known. Specifically, a graph topology inference (GTI) step is employed to select a feasible graph topology, i.e., one having the desired property. Then, a graph weight estimation (GWE) step is performed by solving a generalized graph Laplacian estimation problem, where edges are constrained by the topology found in the GTI step. Our main result is a bound on the error of the GWE step as a function of the error in the GTI step. This error bound indicates that the GTI step should be solved using an algorithm that approximates the similarity matrix by another matrix whose entries have been thresholded to zero to have the desired type of graph topology. The GTI stage can leverage existing methods (e.g., state of the art approaches for graph coloring) which are typically based on minimizing the total weight of removed edges. Since the GWE stage is formulated as an inverse covariance estimation problem with linear constraints, it can be solved using existing convex optimization methods. We demonstrate that our two step approach can achieve good results for both synthetic and texture image data.
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