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Register a new userFinite $3$-connected homogeneous graphs
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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.
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A multigraph is exactly k-edge-connected if there are exactly k edge-disjoint paths between any pair of vertices. We characterize the class of exactly 3-edge-connected graphs, giving a synthesis involving two operations by which every exactly 3-edge-connected multigraph can be generated. Slightly modified syntheses give the planar exactly 3-edge-connected graphs and the exactly 3-edge-connected graphs with the fewest possible edges.
A graph $Gamma$ is $k$-connected-homogeneous ($k$-CH) if $k$ is a positive integer and any isomorphism between connected induced subgraphs of order at most $k$ extends to an automorphism of $Gamma$, and connected-homogeneous (CH) if this property holds for all $k$. Locally finite, locally connected graphs often fail to be 4-CH because of a combinatorial obstruction called the unique $x$ property; we prove that this property holds for locally strongly regular graphs under various purely combinatorial assumptions. We then classify the locally finite, locally connected 4-CH graphs. We also classify the locally finite, locally disconnected 4-CH graphs containing 3-cycles and induced 4-cycles, and prove that, with the possible exception of locally disconnected graphs containing 3-cycles but no induced 4-cycles, every finite 7-CH graph is CH.
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 $d_{0}(Q_{k,r})=k+r={Gamma}(G)+{gamma}(G)$.
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.
A $k$-connected set in an infinite graph, where $k > 0$ is an integer, is a set of vertices such that any two of its subsets of the same size $ell leq k$ can be connected by $ell$ disjoint paths in the whole graph. We characterise the existence of $k$-connected sets of arbitrary but fixed infinite cardinality via the existence of certain minors and topological minors. We also prove a duality theorem for the existence of such $k$-connected sets: if a graph contains no such $k$-connected set, then it has a tree-decomposition which, whenever it exists, precludes the existence of such a $k$-connected set.
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