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We say that a graph $G$ is $(2,m)$-linked if, for any distinct vertices $a_1,\ldots, a_m, b_1,b_2$ in $G$, there exist vertex disjoint connected subgraphs $A,B$ of $G$ such that $\{a_1, \ldots, a_m\}$ is contained in $A$ and $\{b_1,b_2\}$ is contained in $B$. A fundamental result in structural graph theory is the characterization of $(2,2)$-linked graphs, with different versions obtained independently by Robertson and Chakravarty, Seymour, and Thomassen. It appears to be very difficult to characterize $(2,m)$-linked graphs for $m\ge 3$. In this paper, we provide a partial characterization of $(2,m)$-linked graphs by adding an average degree condition. This implies that $(2m+2)$-connected graphs are $(2,m)$-linked. Moreover, if $G$ is a $(2m+2)$-connected graph and $a_1, \ldots, a_m, b_1,b_2$ are distinct vertices of $G$, then there is a path $P$ in $G$ between $b_1$ and $b_2$ and avoiding $\{a_1, \ldots, a_m\}$ such that $G-P$ is connected, improving a previous connectivity bound of $10m$., Comment: 17 pages

We prove that every $P_5$-free graph of bounded clique number contains a small hitting set of all its maximum stable sets. More generally, let us say a class $\mathcal{C}$ of graphs is $\eta$-bounded if there exists a function $h:\mathbb{N}\rightarrow \mathbb{N}$ such that $\eta(G)\leq h(\omega(G))$ for every graph $G\in \mathcal{C}$, where $\eta(G)$ denotes smallest cardinality of a hitting set of all maximum stable sets in $G$, and $\omega(G)$ is the clique number of $G$. Also, $\mathcal{C}$ is said to be polynomially $\eta$-bounded if in addition $h$ can be chosen to be a polynomial. We introduce $\eta$-boundedness inspired by a question of Alon and motivated by a number of meaningful similarities to $\chi$-boundedness. In particular, we propose an analogue of the Gy\'{a}rf\'{a}s-Sumner conjecture, that the class of all $H$-free graphs is $\eta$-bounded if (and only if) $H$ is a forest. Like $\chi$-boundedness, the case where $H$ is a star is easy to verify, and we prove two non-trivial extensions of this: $H$-free graphs are $\eta$-bounded if (1) $H$ has a vertex incident with all edges of $H$, or (2) $H$ can be obtained from a star by subdividing at most one edge, exactly once. Unlike $\chi$-boundedness, the case where $H$ is a path is surprisingly hard. Our main result mentioned at the beginning shows that $P_5$-free graphs are $\eta$-bounded. The proof is rather involved compared to the classical ``Gy\'{a}rf\'{a}s path'' argument which establishes, for all $t$, the $\chi$-boundedness of $P_t$-free graphs. It remains open whether $P_t$-free graphs are $\eta$-bounded for $t\geq 6$. It also remains open whether $P_5$-free graphs are polynomially $\eta$-bounded, which, if true, would imply the Erd\H{o}s-Hajnal conjecture for $P_5$-free graphs. But we prove that $H$-free graphs are polynomially $\eta$-bounded if $H$ is a proper induced subgraph of $P_5$., Comment: [v2] Accepted manuscript; see DOI for journal version [v3] Fixed mistake in (16)

The List-3-Coloring Problem is to decide, given a graph $G$ and a list $L(v)\subseteq \{1,2,3\}$ of colors assigned to each vertex $v$ of $G$, whether $G$ admits a proper coloring $\phi$ with $\phi(v)\in L(v)$ for every vertex $v$ of $G$, and the $3$-Coloring Problem is the List-$3$-Coloring Problem on instances with $L(v)=\{1,2,3\}$ for every vertex $v$ of $G$. The List-$3$-Coloring Problem is a classical NP-complete problem, and it is well-known that while restricted to $H$-free graphs (meaning graphs with no induced subgraph isomorphic to a fixed graph $H$), it remains NP-complete unless $H$ is isomorphic to an induced subgraph of a path. However, the current state of art is far from proving this to be sufficient for a polynomial time algorithm; in fact, the complexity of the $3$-Coloring Problem on $P_8$-free graphs (where $P_8$ denotes the eight-vertex path) is unknown. Here we consider a variant of the List-$3$-Coloring Problem called the Ordered Graph List-$3$-Coloring Problem, where the input is an ordered graph, that is, a graph along with a linear order on its vertex set. For ordered graphs $G$ and $H$, we say $G$ is $H$-free if $H$ is not isomorphic to an induced subgraph of $G$ with the isomorphism preserving the linear order. We prove, assuming $H$ to be an ordered graph, a nearly complete dichotomy for the Ordered Graph List-$3$-Coloring Problem restricted to $H$-free ordered graphs. In particular, we show that the problem can be solved in polynomial time if $H$ has at most one edge, and remains NP-complete if $H$ has at least three edges. Moreover, in the case where $H$ has exactly two edges, we give a complete dichotomy when the two edges of $H$ share an end, and prove several NP-completeness results when the two edges of $H$ do not share an end, narrowing the open cases down to three very special types of two-edge ordered graphs., Comment: Accepted manuscript; see DOI for journal version

The dissemination of fake news on social networks has drawn public need for effective and efficient fake news detection methods. Generally, fake news on social networks is multi-modal and has various connections with other entities such as users and posts. The heterogeneity in both news content and the relationship with other entities in social networks brings challenges to designing a model that comprehensively captures the local multi-modal semantics of entities in social networks and the global structural representation of the propagation patterns, so as to classify fake news effectively and accurately. In this paper, we propose a novel Transformer-based model: HetTransformer to solve the fake news detection problem on social networks, which utilises the encoder-decoder structure of Transformer to capture the structural information of news propagation patterns. We first capture the local heterogeneous semantics of news, post, and user entities in social networks. Then, we apply Transformer to capture the global structural representation of the propagation patterns in social networks for fake news detection. Experiments on three real-world datasets demonstrate that our model is able to outperform the state-of-the-art baselines in fake news detection.

Motivated by the analogous questions in graphs, we study the complexity of coloring and stable set problems in hypergraphs with forbidden substructures and bounded edge size. Letting $\nu(G)$ denote the maximum size of a matching in $H$, we obtain complete dichotomies for the complexity of the following problems parametrized by fixed $r, k, s \in \mathbb{N}$: $r$-Coloring in hypergraphs $G$ with edge size at most $k$ and $\nu(G) \leq s$; $r$-Precoloring Extension in $k$-uniform hypergraphs $G$ with $\nu(G) \leq s$; $r$-Precoloring Extension in hypergraphs $G$ with edge size at most $k$ and $\nu(G) \leq s$; Maximum Stable Set in $k$-uniform hypergraphs $G$ with $\nu(G) \leq s$; Maximum Weight Stable Set in $k$-uniform hypergraphs with $\nu(G) \leq s$; as well as partial results for $r$-Coloring in $k$-uniform hypergraphs $\nu(G) \leq s$. We then turn our attention to $2$-Coloring in 3-uniform hypergraphs with forbidden induced subhypergraphs, and give a polynomial-time algorithm when restricting the input to hypergraphs excluding a fixed one-edge hypergraph. Finally, we consider linear 3-uniform hypergraphs (in which every two edges share at most one vertex), and show that excluding an induced matching in $G$ implies that $\nu(G)$ is bounded by a constant; and that $3$-coloring linear $3$-uniform hypergraphs $G$ with $\nu(G) \leq 532$ is NP-hard., Comment: Accepted manuscript; see DOI for journal version

For a positive integer $r$ and graphs $G$ and $H$, we denote by $G+H$ the disjoint union of $G$ and $H$, and by $rH$ the union of $r$ mutually disjoint copies of $H$. Also, we say $G$ is $H$-free if $H$ is not isomorphic to an induced subgraph of $G$. We use $P_t$ to denote the path on $t$ vertices. For a fixed positive integer $k$, the List-$k$-Coloring Problem is to decide, given a graph $G$ and a list $L(v)\subseteq \{1,\ldots,k\}$ of colors assigned to each vertex $v$ of $G$, whether $G$ admits a proper coloring $\phi$ with $\phi(v)\in L(v)$ for every vertex $v$ of $G$, and the $k$-Coloring Problem is the List-$k$-Coloring Problem restricted to instances with $L(v)=\{1,\ldots, k\}$ for every vertex $v$ of $G$. We prove that for every positive integer $r$, the List-$5$-Coloring Problem restricted to $rP_3$-free graphs can be solved in polynomial time. Together with known results, this gives a complete dichotomy for the complexity of the List-$5$-Coloring Problem restricted to $H$-free graphs: For every graph $H$, assuming P$\neq$NP, the List-$5$-Coloring Problem restricted to $H$-free graphs can be solved in polynomial time if and only if $H$ is an induced subgraph of either $rP_3$ or $P_5+rP_1$ for some positive integer $r$. As a hardness counterpart, we also show that the $k$-Coloring Problem restricted to $rP_4$-free graphs is NP-complete for all $k\geq 5$ and $r\geq 2$., Comment: Accepted manuscript, see DOI for journal version

Achieving higher sensitivity is an earnest purpose for precision metrology. As a response to this goal, the weak value amplification approach has been developed for measuring ultra-small physical effects, realizing sensitivity that had never been reached before. Encouraged by the successes, many efforts have been devoted to obtain ultimate sensitivity of weak value amplification. However, the benefit would be easily compromised in practice, because the cost of significant reduction on signal intensity leads to an ultra-low signal-to-noise ratio. In this work, we bridge this gap by proposing an alternative weak value amplification approach, which provides sensitivity several orders of magnitude higher than the standard approach while being compatible with practical imperfections. In the proof-of-principle experiment of measuring longitudinal phase change in time-domain, sensitivity up to $5\times10^{-4}$ attosecond is exemplified. Our approach can be applied to measure other small parameters with extremely high sensitivity, providing a new method for future precision metrology., Comment: 6 pages, 4 figures. Accepted by Phys.Rev.A