A graph $G$ is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function $w$ is defined on its vertices. Then $G$ is $w$-well-covered if all maximal independent sets are of the same weight. For every graph $G$, the set of weight functions $w$ such that $G$ is $w$-well-covered is a vector space, denoted as $WCW(G).$ Deciding whether an input graph $G$ is well-covered is co-NP-complete. Therefore, finding $WCW(G)$ is co-NP-hard. A generating subgraph of a graph $G$ is an induced complete bipartite subgraph $B$ of $G$ on vertex sets of bipartition $B_{X}$ and $B_{Y}$, such that each of $S \cup B_{X}$ and $S \cup B_{Y}$ is a maximal independent set of $G$, for some independent set $S$. If $B$ is generating, then $w(B_{X})=w(B_{Y})$ for every weight function $w \in WCW(G)$. Therefore, generating subgraphs play an important role in finding $WCW(G)$. The decision problem whether a subgraph of an input graph is generating is known to be NP-complete. In this article, we prove NP-completeness of the problem for graphs without cycles of length 3 and 5, and for bipartite graphs with girth at least 6. On the other and, we supply polynomial algorithms for recognizing generating subgraphs and finding $WCW(G)$, when the input graph is bipartite without cycles of length 6. We also present a polynomial algorithm which finds $WCW(G)$ when $G$ does not contain cycles of lengths 3, 4, 5, and 7.
Let $B$ be an induced complete bipartite subgraph of $G$ on vertex sets of bipartition $B_{X}$ and $B_{Y}$. The subgraph $B$ is {\it generating} if there exists an independent set $S$ such that each of $S \cup B_{X}$ and $S \cup B_{Y}$ is a maximal independent set in the graph. If $B$ is generating, it \textit{produces} the restriction $w(B_{X})=w(B_{Y})$. Let $w:V(G) \longrightarrow\mathbb{R}$ be a weight function. We say that $G$ is $w$-well-covered if all maximal independent sets are of the same weight. The graph $G$ is $w$-well-covered if and only if $w$ satisfies all restrictions produced…
A bipartite graph $G=(L,R;E)$ with at least one edge is said to be identifiable if for every vertex $v\in L$, the subgraph induced by its non-neighbors has a matching of cardinality $|L|-1$. An $\ell$-subgraph of $G$ is an induced subgraph of $G$ obtained by deleting from it some vertices in $L$ together with all their neighbors. The Identifiable Subgraph problem is the problem of determining whether a given bipartite graph contains an identifiable $\ell$-subgraph. We show that the Identifiable Subgraph problem is polynomially solvable, along with the version of the problem in which the task i…
A simple-triangle graph (also known as a PI graph) is the intersection graph of a family of triangles defined by a point on a horizontal line and an interval on another horizontal line. The recognition problem for simple-triangle graphs was a longstanding open problem, and recently a polynomial-time algorithm has been given [G. B. Mertzios, The Recognition of Simple-Triangle Graphs and of Linear-Interval Orders is Polynomial, SIAM J. Discrete Math., 29(3):1150--1185, 2015]. Along with the approach of this paper, we show a simpler recognition algorithm for simple-triangle graphs. To do this, we…
A graph $G$ is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function $w$ is defined on its vertices. Then $G$ is $w$-well-covered if all maximal independent sets are of the same weight. For every graph $G$, the set of weight functions $w$ such that $G$ is $w$-well-covered is a vector space, denoted $WCW(G)$. Let $B$ be a complete bipartite induced subgraph of $G$ on vertex sets of bipartition $B_{X}$ and $B_{Y}$. Then $B$ is generating if there exists an independent set $S$ such that $S \cup B_{X}$ and $S \cup B_{Y}$ are both maximal indepe…
S-prime graphs are graphs that cannot be represented as nontrivial subgraphs of nontrivial Cartesian products of graphs, i.e., whenever it is a subgraph of a nontrivial Cartesian product graph it is a subgraph of one the factors. A graph is S-composite if it is not S-prime. Although linear time recognition algorithms for determining whether a graph is prime or not with respect to the Cartesian product are known, it remained unknown if a similar result holds also for the recognition of S-prime and S-composite graphs. In this contribution the computational complexity of recognizing S-composite a…
University of Ottawa（渥太华大学）；Jagiellonian University（雅盖隆大学）；Université libre de Bruxelles（布鲁塞尔自由大学）；École Normale Supérieure, CNRS, PSL University（巴黎高等师范学院、法国国家科学研究中心、巴黎文理研究大学）；Carleton University（卡尔顿大学）；Monash University（蒙纳士大学）
A matching-cut of a graph is an edge cut that is a matching. The problem Matching-Cut is that of recognizing graphs with a matching-cut and is NP-complete, even if the graph belongs to one of a number of classes. We initiate the study of Matching-Cut for graphs without a fixed path as an induced subgraph. We show that Matching-Cut is in P for $P_5$-free graphs, but that there exists an integer $t > 0$ for which it is NP-complete for $P_{t}$-free graphs.
We present an efficient algorithm for a problem in the interface between clustering and graph embeddings. An embedding $φ:G\rightarrow M$ of a graph $G$ into a 2-manifold $M$ maps the vertices in $V(G)$ to distinct points and the edges in $E(G)$ to interior-disjoint Jordan arcs between the corresponding vertices. In applications in clustering, cartography, and visualization, nearby vertices and edges are often bundled to the same point or overlapping arcs, due to data compression or low resolution. This raises the computational problem of deciding whether a given map $φ:G\rightarrow M$ comes f…
We investigate the parameterized complexity of the recognition problem for the proper $H$-graphs. The $H$-graphs are the intersection graphs of connected subgraphs of a subdivision of a multigraph $H$, and the properness means that the containment relationship between the representations of the vertices is forbidden. The class of $H$-graphs was introduced as a natural (parameterized) generalization of interval and circular-arc graphs by Biró, Hujter, and Tuza in 1992, and the proper $H$-graphs were introduced by Chaplick et al. in WADS 2019 as a generalization of proper interval and circular-a…
Independent set games are cooperative games defined on graphs, where players are edges and the value of a coalition is the maximum cardinality of independent sets in the subgraph defined by the coalition. In this paper, we investigate the convexity of independent set games, as convex games possess many nice properties both economically and computationally. For independent set games introduced by Deng et al. (Math. Oper. Res., 24:751-766, 1999), we provide a necessary and sufficient characterization for the convexity, i.e., every non-pendant edge is incident to a pendant edge in the underlying …
Threshold graphs are recursive deterministic network models that have been proposed for describing certain economic and social interactions. One drawback of this graph family is that it has limited generative attachment rules. To mitigate this problem, we introduce a new class of graphs termed Paired Threshold (PT) graphs described through vertex weights that govern the existence of edges via two inequalities. One inequality imposes the constraint that the sum of weights of adjacent vertices has to exceed a specified threshold. The second inequality ensures that adjacent vertices have a weight…
The independence gap of a graph was introduced by Ekim et al. (2018) as a measure of how far a graph is from being well-covered. It is defined as the difference between the maximum and minimum size of a maximal independent set. We investigate the independence gap of a graph from structural and algorithmic points of view, with a focus on classes of perfect graphs. Generalizing results on well-covered graphs due to Dean and Zito (1994) and Hujdurović et al. (2018), we express the independence gap of a perfect graph in terms of clique partitions and use this characterization to develop a polynomi…
The Maximum Weight Independent Set (MWIS) problem is a well-known NP-hard problem. For graphs $G_1, G_2$, $G_1+G_2$ denotes the disjoint union of $G_1$ and $G_2$, and for a constant $l \ge 2$, $lG$ denotes the disjoint union of $l$ copies of $G$. A {\em claw} has vertices $a,b,c,d$, and edges $ab,ac,ad$. MWIS can be solved for claw-free graphs in polynomial time; the first two polynomial time algorithms were introduced in 1980 by \cite{Minty1980,Sbihi1980}, then revisited by \cite{NakTam2001}, and recently improved by \cite{FaeOriSta2011,FaeOriSta2014}, and by \cite{NobSas2011,NobSas2015} with…
