On detectable colorings of graphs

summary:Let $G$ be a connected graph of order $n \ge 3$ and let $c\: E(G) \rightarrow \lbrace 1, 2, \ldots , k\rbrace$ be a coloring of the edges of $G$ (where adjacent edges may be colored the same). For each vertex $v$ of $G$, the color code of $v$ with respect to $c$ is the $k$-tuple $c(v) = (a_1, a_2, \cdots , a_k)$, where $a_i$ is the number of edges incident with $v$ that are colored $i$ ($1 \le i \le k$). The coloring $c$ is detectable if distinct vertices have distinct color codes. The detection number $\det (G)$ of $G$ is the minimum positive integer $k$ for which $G$ has a detectable $k$-coloring. We establish a formula for the detection number of a path in terms of its order. For each integer $n \ge 3$, let $D_u(n)$ be the maximum detection number among all unicyclic graphs of order $n$ and $d_u(n)$ the minimum detection number among all unicyclic graphs of order $n$. The numbers $D_u(n)$ and $d_u(n)$ are determined for all integers $n \ge 3$. Furthermore, it is shown that for integers $k \ge 2$ and $n \ge 3$, there exists a unicyclic graph $G$ of order $n$ having $\det (G)=k$ if and only if $d_u(n) \le k \le D_u(n)$

The Power of the Weisfeiler-Leman Algorithm to Decompose Graphs

The Weisfeiler-Leman procedure is a widely-used approach for graph isomorphism testing that works by iteratively computing an isomorphism-invariant coloring of vertex tuples. Meanwhile, a fundamental tool in structural graph theory, which is often exploited in approaches to tackle the graph isomorphism problem, is the decomposition into 2- and 3-connected components. We prove that the 2-dimensional Weisfeiler-Leman algorithm implicitly computes the decomposition of a graph into its 3-connected components. Thus, the dimension of the algorithm needed to distinguish two given graphs is at most the dimension required to distinguish the corresponding decompositions into 3-connected components (assuming it is at least 2). This result implies that for k >= 2, the k-dimensional algorithm distinguishes k-separators, i.e., k-tuples of vertices that separate the graph, from other vertex k-tuples. As a byproduct, we also obtain insights about the connectivity of constituent graphs of association schemes. In an application of the results, we show the new upper bound of k on the Weisfeiler-Leman dimension of graphs of treewidth at most k. Using a construction by Cai, F\"urer, and Immerman, we also provide a new lower bound that is asymptotically tight up to a factor of 2.Comment: 30 pages, 4 figures, full version of a paper accepted at MFCS 201

Graphs Identified by Logics with Counting

We classify graphs and, more generally, finite relational structures that are identified by C2, that is, two-variable first-order logic with counting. Using this classification, we show that it can be decided in almost linear time whether a structure is identified by C2. Our classification implies that for every graph identified by this logic, all vertex-colored versions of it are also identified. A similar statement is true for finite relational structures. We provide constructions that solve the inversion problem for finite structures in linear time. This problem has previously been shown to be polynomial time solvable by Martin Otto. For graphs, we conclude that every C2-equivalence class contains a graph whose orbits are exactly the classes of the C2-partition of its vertex set and which has a single automorphism witnessing this fact. For general k, we show that such statements are not true by providing examples of graphs of size linear in k which are identified by C3 but for which the orbit partition is strictly finer than the Ck-partition. We also provide identified graphs which have vertex-colored versions that are not identified by Ck.Comment: 33 pages, 8 Figure