arXiv subjects

Shenwei Huang

Publications and source records attributed to Shenwei Huang.

At least 37 records · Page 2Linked to original sources

arXiv2021

Some Results on $k$-Critical $P_5$-Free Graphs

A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but every proper induced subgraph of $G$ has chromatic number less than $k$. The study of $k$-vertex-critical graphs for graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there is a polynomial-time algorithm to decide if a graph in the class is $(k-1)$-colorable. In this paper, we prove that for every fixed integer $k\ge 1$, there are only finitely many $k$-vertex-critical ($P_5$,gem)-free graphs and $(P_5,\overline{P_3+P_2})$-free graphs. To prove the results we use a known structure theorem for ($P_5$,gem)-free graphs combined with properties of $k$-vertex-critical graphs. Moreover, we characterize all $k$-vertex-critical ($P_5$,gem)-free graphs and $(P_5,\overline{P_3+P_2})$-free graphs for $k \in \{4,5\}$ using a computer generation algorithm.

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arXiv2021

Colouring graphs with no induced six-vertex path or diamond

The diamond is the graph obtained by removing an edge from the complete graph on 4 vertices. A graph is ($P_6$, diamond)-free if it contains no induced subgraph isomorphic to a six-vertex path or a diamond. In this paper we show that the chromatic number of a ($P_6$, diamond)-free graph $G$ is no larger than the maximum of 6 and the clique number of $G$. We do this by reducing the problem to imperfect ($P_6$, diamond)-free graphs via the Strong Perfect Graph Theorem, dividing the imperfect graphs into several cases, and giving a proper colouring for each case. We also show that there is exactly one 6-vertex-critical ($P_6$, diamond, $K_6$)-free graph. Together with the Lovász theta function, this gives a polynomial time algorithm to compute the chromatic number of ($P_6$, diamond)-free graphs.

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arXiv2020

Some extremal results on the chromatic-stability index

The $χ$-stability index ${\rm es}_χ(G)$ of a graph $G$ is the minimum number of its edges whose removal results in a graph with the chromatic number smaller than that of $G$. In this paper three open problems from [European J.\ Combin.\ 84 (2020) 103042] are considered. Examples are constructed which demonstrate that a known characterization of $k$-regular ($k\le 5$) graphs $G$ with ${\rm es}_χ(G) = 1$ does not extend to $k\ge 6$. Graphs $G$ with $χ(G)=3$ for which ${\rm es}_χ(G)+{\rm es}_χ(\overline{G}) = 2$ holds are characterized. Necessary conditions on graphs $G$ which attain a known upper bound on ${\rm es}_χ(G)$ in terms of the order and the chromatic number of $G$ are derived. The conditions are proved to be sufficient when $n\equiv 2 \pmod 3$ and $χ(G)=3$.

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arXiv2020

$k$-Critical Graphs in $P_5$-Free Graphs

Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ be the path on $t$ vertices. A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but every proper induced subgraph of $G$ has chromatic number less than $k$. The study of $k$-vertex-critical graphs for graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there is a polynomial-time algorithm to decide if a graph in the class is $(k-1)$-colorable. In this paper, we initiate a systematic study of the finiteness of $k$-vertex-critical graphs in subclasses of $P_5$-free graphs. Our main result is a complete classification of the finiteness of $k$-vertex-critical graphs in the class of $(P_5,H)$-free graphs for all graphs $H$ on 4 vertices. To obtain the complete dichotomy, we prove the finiteness for four new graphs $H$ using various techniques -- such as Ramsey-type arguments and the dual of Dilworth's Theorem -- that may be of independent interest.

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arXiv2019

A Pvalue-guided Anomaly Detection Approach Combining Multiple Heterogeneous Log Parser Algorithms on IIoT Systems

Industrial Internet of Things (IIoT) is becoming an attack target of advanced persistent threat (APT). Currently, IIoT logs have not been effectively used for anomaly detection. In this paper, we use blockchain to prevent logs from being tampered with and propose a pvalue-guided anomaly detection approach. This approach uses statistical pvalues to combine multiple heterogeneous log parser algorithms. The weighted edit distance is selected as a score function to calculate the nonconformity score between a log and a predefined event. The pvalue is calculated based on the non-conformity scores which indicate how well a log matches an event. This approach is tested on a large number of real-world HDFS logs and IIoT logs. The experiment results show that abnormal events could be effectively recognized by our pvalue-guided approach.

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arXiv2019

On graphs with no induced five-vertex path or paraglider

Given two graphs $H_1$ and $H_2$, a graph is $(H_1,\,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. For a positive integer $t$, $P_t$ is the chordless path on $t$ vertices. A paraglider is the graph that consists of a chorless cycle $C_4$ plus a vertex adjacent to three vertices of the $C_4$. In this paper, we study the structure of ($P_5$, paraglider)-free graphs, and show that every such graph $G$ satisfies $χ(G)\le \lceil \frac{3}{2}ω(G) \rceil$, where $χ(G)$ and $ω(G)$ are the chromatic number and clique number of $G$, respectively. Our bound is attained by the complement of the Clebsch graph on 16 vertices. More strongly, we completely characterize all the ($P_5$, paraglider)-free graphs $G$ that satisfies $χ(G)> \frac{3}{2}ω(G)$. We also construct an infinite family of ($P_5$, paraglider)-free graphs such that every graph $G$ in the family has $χ(G)=\lceil \frac{3}{2}ω(G) \rceil-1$. This shows that our upper bound is optimal up to an additive constant and that there is no $(\frac{3}{2}-ε)$-approximation algorithm to the chromatic number of ($P_5$, paraglider)-free graphs for any $ε>0$.

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arXiv2019

Linearly $χ$-Bounding $(P_6,C_4)$-Free Graphs

Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ and $C_s$ be the path on $t$ vertices and the cycle on $s$ vertices, respectively. In this paper we show that for any $(P_6,C_4)$-free graph $G$ it holds that $χ(G)\le \frac{3}{2}ω(G)$, where $χ(G)$ and $ω(G)$ are the chromatic number and clique number of $G$, respectively. %Our bound is attained by $C_5$ and the Petersen graph. Our bound is attained by several graphs, for instance, the five-cycle, the Petersen graph, the Petersen graph with an additional universal vertex, and all $4$-critical $(P_6,C_4)$-free graphs other than $K_4$ (see \cite{HH17}). The new result unifies previously known results on the existence of linear $χ$-binding functions for several graph classes. Our proof is based on a novel structure theorem on $(P_6,C_4)$-free graphs that do not contain clique cutsets. Using this structure theorem we also design a polynomial time $3/2$-approximation algorithm for coloring $(P_6,C_4)$-free graphs. Our algorithm computes a coloring with $\frac{3}{2}ω(G)$ colors for any $(P_6,C_4)$-free graph $G$ in $O(n^2m)$ time.

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arXiv2018

$(2P_2,K_4)$-Free Graphs are 4-Colorable

In this paper, we show that every $(2P_2,K_4)$-free graph is 4-colorable. The bound is attained by the five-wheel and the complement of the seven-cycle. This answers an open question by Wagon \cite{Wa80} in the 1980s. Our result can also be viewed as a result in the study of the Vizing bound for graph classes. A major open problem in the study of computational complexity of graph coloring is whether coloring can be solved in polynomial time for $(4P_1,C_4)$-free graphs. Lozin and Malyshev \cite{LM17} conjecture that the answer is yes. As an application of our main result, we provide the first positive evidence to the conjecture by giving a 2-approximation algorithm for coloring $(4P_1,C_4)$-free graphs.

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arXiv2018

An Optimal $χ$-Bound for ($P_6$, diamond)-Free Graphs

Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ be the path on $t$ vertices and $K_t$ be the complete graph on $t$ vertices. The diamond is the graph obtained from $K_4$ by removing an edge. In this paper we show that every ($P_6$, diamond)-free graph $G$ satisfies $χ(G)\le ω(G)+3$, where $χ(G)$ and $ω(G)$ are the chromatic number and clique number of $G$, respectively. Our bound is attained by the complement of the famous 27-vertex Schläfli graph. Our result unifies previously known results on the existence of linear $χ$-binding functions for several graph classes. Our proof is based on a reduction via the Strong Perfect Graph Theorem to imperfect ($P_6$, diamond)-free graphs, a careful analysis of the structure of those graphs, and a computer search that relies on a well-known characterization of 3-colourable $(P_6,K_3)$-free graphs.

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arXiv2018

List-three-coloring graphs with no induced $P_6+rP_3$

For an integer $r$, the graph $P_6+rP_3$ has $r+1$ components, one of which is a path on $6$ vertices, and each of the others is a path on $3$ vertices. In this paper we provide a polynomial-time algorithm to test if a graph with no induced subgraph isomorphic to $P_6+rP_3$ is three-colorable. We also solve the list version of this problem, where each vertex is assigned a list of possible colors, which is a subset of $\{1,2,3\}$.

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arXiv2018

Colouring Square-Free Graphs without Long Induced Paths

The complexity of {\sc Colouring} is fully understood for $H$-free graphs, but there are still major complexity gaps if two induced subgraphs $H_1$ and $H_2$ are forbidden. Let $H_1$ be the $s$-vertex cycle $C_s$ and $H_2$ be the $t$-vertex path $P_t$. We show that {\sc Colouring} is polynomial-time solvable for $s=4$ and $t\leq 6$, strengthening several known results. Our main approach is to initiate a study into the boundedness of the clique-width of atoms (graphs with no clique cutset) of a hereditary graph class. We first show that the classifications of boundedness of clique-width of $H$-free graphs and $H$-free atoms coincide. We then show that this is not the case if two graphs are forbidden: we prove that $(C_4,P_6)$-free atoms have clique-width at most~18. Our key proof ingredients are a divide-and-conquer approach for bounding the clique-width of a subclass of $C_4$-free graphs and the construction of a new bound on the clique-width for (general) graphs in terms of the clique-width of recursively defined subgraphs induced by homogeneous pairs and triples of sets. As a complementary result we prove that {\sc Colouring} is \NP-complete for $s=4$ and $t\geq 9$, which is the first hardness result on {\sc Colouring} for $(C_4,P_t)$-free graphs. Combining our new results with known results leads to an almost complete dichotomy for \cn restricted to $(C_s,P_t)$-free graphs.

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arXiv2018

The class of $(P_7,C_4,C_5)$-free graphs: decomposition, algorithms, and $χ$-boundedness

As usual, $P_n$ ($n \geq 1$) denotes the path on $n$ vertices, and $C_n$ ($n \geq 3$) denotes the cycle on $n$ vertices. For a family $\mathcal{H}$ of graphs, we say that a graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to any graph in $\mathcal{H}$. We present a decomposition theorem for the class of $(P_7,C_4,C_5)$-free graphs; in fact, we give a complete structural characterization of $(P_7,C_4,C_5)$-free graphs that do not admit a clique-cutset. We use this decomposition theorem to show that the class of $(P_7,C_4,C_5)$-free graphs is $χ$-bounded by a linear function (more precisely, every $(P_7,C_4,C_5)$-free graph $G$ satisfies $χ(G) \leq \frac{3}{2} ω(G)$). We also use the decomposition theorem to construct an $O(n^3)$ algorithm for the minimum coloring problem, an $O(n^2m)$ algorithm for the maximum weight stable set problem, and an $O(n^3)$ algorithm for the maximum weight clique problem for this class, where $n$ denotes the number of vertices and $m$ the number of edges of the input graph.

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arXiv2017

A Polynomial-time Algorithm to Achieve Extended Justified Representation

We consider a committee voting setting in which each voter approves of a subset of candidates and based on the approvals, a target number of candidates are to be selected. In particular we focus on the axiomatic property called extended justified representation (EJR). Although a committee satisfying EJR is guaranteed to exist, the computational complexity of finding such a committee has been an open problem and explicitly mentioned in multiple recent papers. We settle the complexity of finding a committee satisfying EJR by presenting a polynomial-time algorithm for the problem. Our algorithmic approach may be useful for constructing other voting rules in multi-winner voting.

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arXiv2017

Computational Complexity of Testing Proportional Justified Representation

We consider a committee voting setting in which each voter approves of a subset of candidates and based on the approvals, a target number of candidates are selected. Aziz et al. (2015) proposed two representation axioms called justified representation and extended justified representation. Whereas the former can be tested as well as achieved in polynomial time, the latter property is coNP-complete to test and no polynomial-time algorithm is known to achieve it. Interestingly, S{á}nchez-Fern{á}ndez et~al. (2016) proposed an intermediate property called proportional justified representation that admits a polynomial-time algorithm to achieve. The complexity of testing proportional justified representation has remained an open problem. In this paper, we settle the complexity by proving that testing proportional justified representation is coNP-complete. We complement the complexity result by showing that the problem admits efficient algorithms if any of the following parameters are bounded: (1) number of voters (2) number of candidates (3) maximum number of candidates approved by a voter (4) maximum number of voters approving a given candidate.

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arXiv2016

Structure and algorithms for (cap, even hole)-free graphs

A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free graphs, and more generally, (cap, 4-hole)-free odd-signable graphs. We give an explicit construction of these graphs. We prove that every such graph $G$ has a vertex of degree at most $\frac{3}{2}ω(G) -1$, and hence $χ(G)\leq \frac{3}{2}ω(G)$, where $ω(G)$ denotes the size of a largest clique in $G$ and $χ(G)$ denotes the chromatic number of $G$. We give an $O(nm)$ algorithm for $q$-coloring these graphs for fixed $q$ and an $O(nm)$ algorithm for maximum weight stable set. We also give a polynomial-time algorithm for minimum coloring. Our algorithms are based on our results that triangle-free odd-signable graphs have treewidth at most 5 and thus have clique-width at most 48, and that (cap, 4-hole)-free odd-signable graphs $G$ without clique cutsets have treewidth at most $6ω(G)-1$ and clique-width at most 48.

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arXiv2015

A note on coloring (even-hole,cap)-free graphs

A {\em hole} is a chordless cycle of length at least four. A hole is {\em even} (resp. {\em odd}) if it contains an even (resp. odd) number of vertices. A \emph{cap} is a graph induced by a hole with an additional vertex that is adjacent to exactly two adjacent vertices on the hole. In this note, we use a decomposition theorem by Conforti et al. (1999) to show that if a graph $G$ does not contain any even hole or cap as an induced subgraph, then $χ(G)\le \lfloor\frac{3}{2}ω(G)\rfloor$, where $χ(G)$ and $ω(G)$ are the chromatic number and the clique number of $G$, respectively. This bound is attained by odd holes and the Hajos graph. The proof leads to a polynomial-time $3/2$-approximation algorithm for coloring (even-hole,cap)-free graphs.

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arXiv2015

Bounding the Clique-Width of $H$-free Chordal Graphs

A graph is $H$-free if it has no induced subgraph isomorphic to $H$. Brandstädt, Engelfriet, Le and Lozin proved that the class of chordal graphs with independence number at most 3 has unbounded clique-width. Brandstädt, Le and Mosca erroneously claimed that the gem and the co-gem are the only two 1-vertex $P_4$-extensions $H$ for which the class of $H$-free chordal graphs has bounded clique-width. In fact we prove that bull-free chordal and co-chair-free chordal graphs have clique-width at most 3 and 4, respectively. In particular, we find four new classes of $H$-free chordal graphs of bounded clique-width. Our main result, obtained by combining new and known results, provides a classification of all but two stubborn cases, that is, with two potential exceptions we determine all graphs $H$ for which the class of $H$-free chordal graphs has bounded clique-width. We illustrate the usefulness of this classification for classifying other types of graph classes by proving that the class of $(2P_1+P_3,K_4)$-free graphs has bounded clique-width via a reduction to $K_4$-free chordal graphs. Finally, we give a complete classification of the (un)boundedness of clique-width of $H$-free weakly chordal graphs.

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arXiv2015

Bounding the Clique-Width of $H$-free Split Graphs

A graph is $H$-free if it has no induced subgraph isomorphic to $H$. We continue a study into the boundedness of clique-width of subclasses of perfect graphs. We identify five new classes of $H$-free split graphs whose clique-width is bounded. Our main result, obtained by combining new and known results, provides a classification of all but two stubborn cases, that is, with two potential exceptions we determine all graphs $H$ for which the class of $H$-free split graphs has bounded clique-width.

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