A dominating set $D$ of a graph $G$ without isolated vertices is called semipaired dominating set if $D$ can be partitioned into $2$-element subsets such that the vertices in each set are at distance at most $2$. The semipaired domination number, denoted by $gamma_{pr2}(G)$ is the minimum cardinality of a semipaired dominating set of $G$. Given a graph $G$ with no isolated vertices, the textsc{Minimum Semipaired Domination} problem is to find a semipaired dominating set of $G$ of cardinality $gamma_{pr2}(G)$. The decision version of the textsc{Minimum Semipaired Domination} problem is already known to be NP-complete for chordal graphs, an important graph class. In this paper, we show that the decision version of the textsc{Minimum Semipaired Domination} problem remains NP-complete for split graphs, a subclass of chordal graphs. On the positive side, we propose a linear-time algorithm to compute a minimum cardinality semipaired dominating set of block graphs. In addition, we prove that the textsc{Minimum Semipaired Domination} problem is APX-complete for graphs with maximum degree $3$.
{"title":"Semipaired Domination in Some Subclasses of Chordal Graphs","authors":"Michael A. Henning, Arti Pandey, Vikash Tripathi","doi":"10.46298/dmtcs.6782","DOIUrl":"https://doi.org/10.46298/dmtcs.6782","url":null,"abstract":"A dominating set $D$ of a graph $G$ without isolated vertices is called\u0000semipaired dominating set if $D$ can be partitioned into $2$-element subsets\u0000such that the vertices in each set are at distance at most $2$. The semipaired\u0000domination number, denoted by $gamma_{pr2}(G)$ is the minimum cardinality of a\u0000semipaired dominating set of $G$. Given a graph $G$ with no isolated vertices,\u0000the textsc{Minimum Semipaired Domination} problem is to find a semipaired\u0000dominating set of $G$ of cardinality $gamma_{pr2}(G)$. The decision version of\u0000the textsc{Minimum Semipaired Domination} problem is already known to be\u0000NP-complete for chordal graphs, an important graph class. In this paper, we\u0000show that the decision version of the textsc{Minimum Semipaired Domination}\u0000problem remains NP-complete for split graphs, a subclass of chordal graphs. On\u0000the positive side, we propose a linear-time algorithm to compute a minimum\u0000cardinality semipaired dominating set of block graphs. In addition, we prove\u0000that the textsc{Minimum Semipaired Domination} problem is APX-complete for\u0000graphs with maximum degree $3$.\u0000","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114011882","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article we present theoretical and computational results on the existence of polyhedral embeddings of graphs. The emphasis is on cubic graphs. We also describe an efficient algorithm to compute all polyhedral embeddings of a given cubic graph and constructions for cubic graphs with some special properties of their polyhedral embeddings. Some key results are that even cubic graphs with a polyhedral embedding on the torus can also have polyhedral embeddings in arbitrarily high genus, in fact in a genus {em close} to the theoretical maximum for that number of vertices, and that there is no bound on the number of genera in which a cubic graph can have a polyhedral embedding. While these results suggest a large variety of polyhedral embeddings, computations for up to 28 vertices suggest that by far most of the cubic graphs do not have a polyhedral embedding in any genus and that the ratio of these graphs is increasing with the number of vertices.
{"title":"On the genera of polyhedral embeddings of cubic graph","authors":"G. Brinkmann, T. Tucker, N. Cleemput","doi":"10.46298/dmtcs.6729","DOIUrl":"https://doi.org/10.46298/dmtcs.6729","url":null,"abstract":"In this article we present theoretical and computational results on the existence of polyhedral embeddings of graphs. The emphasis is on cubic graphs. We also describe an efficient algorithm to compute all polyhedral embeddings of a given cubic graph and constructions for cubic graphs with some special properties of their polyhedral embeddings. Some key results are that even cubic graphs with a polyhedral embedding on the torus can also have polyhedral embeddings in arbitrarily high genus, in fact in a genus {em close} to the theoretical maximum for that number of vertices, and that there is no bound on the number of genera in which a cubic graph can have a polyhedral embedding. While these results suggest a large variety of polyhedral embeddings, computations for up to 28 vertices suggest that by far most of the cubic graphs do not have a polyhedral embedding in any genus and that the ratio of these graphs is increasing with the number of vertices.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125970518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider nondeterministic higher-order recursion schemes as recognizers of languages of finite words or finite trees. We propose a type system that allows to solve the simultaneous-unboundedness problem (SUP) for schemes, which asks, given a set of letters A and a scheme G, whether it is the case that for every number n the scheme accepts a word (a tree) in which every letter from A appears at least n times. Using this type system we prove that SUP is (m-1)-EXPTIME-complete for word-recognizing schemes of order m, and m-EXPTIME-complete for tree-recognizing schemes of order m. Moreover, we establish the reflection property for SUP: out of an input scheme G one can create its enhanced version that recognizes the same language but is aware of the answer to SUP.
{"title":"A Type System Describing Unboundedness","authors":"P. Parys","doi":"10.23638/DMTCS-22-4-2","DOIUrl":"https://doi.org/10.23638/DMTCS-22-4-2","url":null,"abstract":"We consider nondeterministic higher-order recursion schemes as recognizers of languages of finite words or finite trees. We propose a type system that allows to solve the simultaneous-unboundedness problem (SUP) for schemes, which asks, given a set of letters A and a scheme G, whether it is the case that for every number n the scheme accepts a word (a tree) in which every letter from A appears at least n times. Using this type system we prove that SUP is (m-1)-EXPTIME-complete for word-recognizing schemes of order m, and m-EXPTIME-complete for tree-recognizing schemes of order m. Moreover, we establish the reflection property for SUP: out of an input scheme G one can create its enhanced version that recognizes the same language but is aware of the answer to SUP.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"170 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115273716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Hamiltonian cycles in graphs were first studied in the 1850s. Since then, an impressive amount of research has been dedicated to identifying classes of graphs that allow Hamiltonian cycles, and to related questions. The corresponding decision problem, that asks whether a given graph is Hamiltonian (i.,e. admits a Hamiltonian cycle), is one of Karp's famous NP-complete problems. In this paper we study graphs of bounded degree that are emph{far} from being Hamiltonian, where a graph $G$ on $n$ vertices is emph{far} from being Hamiltonian, if modifying a constant fraction of $n$ edges is necessary to make $G$ Hamiltonian. We give an explicit deterministic construction of a class of graphs of bounded degree that are locally Hamiltonian, but (globally) far from being Hamiltonian. Here, emph{locally Hamiltonian} means that every subgraph induced by the neighbourhood of a small vertex set appears in some Hamiltonian graph. More precisely, we obtain graphs which differ in $Theta(n)$ edges from any Hamiltonian graph, but non-Hamiltonicity cannot be detected in the neighbourhood of $o(n)$ vertices. Our class of graphs yields a class of hard instances for one-sided error property testers with linear query complexity. It is known that any property tester (even with two-sided error) requires a linear number of queries to test Hamiltonicity (Yoshida, Ito, 2010). This is proved via a randomised construction of hard instances. In contrast, our construction is deterministic. So far only very few deterministic constructions of hard instances for property testing are known. We believe that our construction may lead to future insights in graph theory and towards a characterisation of the properties that are testable in the bounded-degree model.
{"title":"An explicit construction of graphs of bounded degree that are far from being Hamiltonian","authors":"Isolde Adler, N. Köhler","doi":"10.46298/dmtcs.7109","DOIUrl":"https://doi.org/10.46298/dmtcs.7109","url":null,"abstract":"Hamiltonian cycles in graphs were first studied in the 1850s. Since then, an\u0000impressive amount of research has been dedicated to identifying classes of\u0000graphs that allow Hamiltonian cycles, and to related questions. The\u0000corresponding decision problem, that asks whether a given graph is Hamiltonian\u0000(i.,e. admits a Hamiltonian cycle), is one of Karp's famous NP-complete\u0000problems. In this paper we study graphs of bounded degree that are emph{far}\u0000from being Hamiltonian, where a graph $G$ on $n$ vertices is emph{far} from\u0000being Hamiltonian, if modifying a constant fraction of $n$ edges is necessary\u0000to make $G$ Hamiltonian. We give an explicit deterministic construction of a\u0000class of graphs of bounded degree that are locally Hamiltonian, but (globally)\u0000far from being Hamiltonian. Here, emph{locally Hamiltonian} means that every\u0000subgraph induced by the neighbourhood of a small vertex set appears in some\u0000Hamiltonian graph. More precisely, we obtain graphs which differ in $Theta(n)$\u0000edges from any Hamiltonian graph, but non-Hamiltonicity cannot be detected in\u0000the neighbourhood of $o(n)$ vertices. Our class of graphs yields a class of\u0000hard instances for one-sided error property testers with linear query\u0000complexity. It is known that any property tester (even with two-sided error)\u0000requires a linear number of queries to test Hamiltonicity (Yoshida, Ito, 2010).\u0000This is proved via a randomised construction of hard instances. In contrast,\u0000our construction is deterministic. So far only very few deterministic\u0000constructions of hard instances for property testing are known. We believe that\u0000our construction may lead to future insights in graph theory and towards a\u0000characterisation of the properties that are testable in the bounded-degree\u0000model.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122289739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we present an average-case analysis of closed lambda terms with restricted values of De Bruijn indices in the model where each occurrence of a variable contributes one to the size. Given a fixed integer k, a lambda term in which all De Bruijn indices are bounded by k has the following shape: It starts with k De Bruijn levels, forming the so-called hat of the term, to which some number of k-colored Motzkin trees are attached. By means of analytic combinatorics, we show that the size of this hat is constant on average and that the average number of De Bruijn levels of k-colored Motzkin trees of size n is asymptotically Θ(√ n). Combining these two facts, we conclude that the maximal non-empty De Bruijn level in a lambda term with restrictions on De Bruijn indices and of size n is, on average, also of order √ n. On this basis, we provide the average unary profile of such lambda terms.
{"title":"Unary profile of lambda terms with restricted De Bruijn indices","authors":"Katarzyna Grygiel, Isabella Larcher","doi":"10.46298/dmtcs.5836","DOIUrl":"https://doi.org/10.46298/dmtcs.5836","url":null,"abstract":"\u0000 In this paper we present an average-case analysis of closed lambda terms with restricted values of De Bruijn indices in the model where each occurrence of a variable contributes one to the size. Given a fixed integer k, a lambda term in which all De Bruijn indices are bounded by k has the following shape: It starts with k De Bruijn levels, forming the so-called hat of the term, to which some number of k-colored Motzkin trees are attached. By means of analytic combinatorics, we show that the size of this hat is constant on average and that the average number of De Bruijn levels of k-colored Motzkin trees of size n is asymptotically Θ(√ n). Combining these two facts, we conclude that the maximal non-empty De Bruijn level in a lambda term with restrictions on De Bruijn indices and of size n is, on average, also of order √ n. On this basis, we provide the average unary profile of such lambda terms.\u0000","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116838155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk
The number of down-steps between pairs of up-steps in $k_t$-Dyck paths, a generalization of Dyck paths consisting of steps ${(1, k), (1, -1)}$ such that the path stays (weakly) above the line $y=-t$, is studied. Results are proved bijectively and by means of generating functions, and lead to several interesting identities as well as links to other combinatorial structures. In particular, there is a connection between $k_t$-Dyck paths and perforation patterns for punctured convolutional codes (binary matrices) used in coding theory. Surprisingly, upon restriction to usual Dyck paths this yields a new combinatorial interpretation of Catalan numbers.
{"title":"Down-step statistics in generalized Dyck paths","authors":"Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk","doi":"10.46298/dmtcs.7163","DOIUrl":"https://doi.org/10.46298/dmtcs.7163","url":null,"abstract":"The number of down-steps between pairs of up-steps in $k_t$-Dyck paths, a\u0000generalization of Dyck paths consisting of steps ${(1, k), (1, -1)}$ such\u0000that the path stays (weakly) above the line $y=-t$, is studied. Results are\u0000proved bijectively and by means of generating functions, and lead to several\u0000interesting identities as well as links to other combinatorial structures. In\u0000particular, there is a connection between $k_t$-Dyck paths and perforation\u0000patterns for punctured convolutional codes (binary matrices) used in coding\u0000theory. Surprisingly, upon restriction to usual Dyck paths this yields a new\u0000combinatorial interpretation of Catalan numbers.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127416088","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $P$ be a set of $ngeq 3$ points in general position in the plane. The edge disjointness graph $D(P)$ of $P$ is the graph whose vertices are all the closed straight line segments with endpoints in $P$, two of which are adjacent in $D(P)$ if and only if they are disjoint. We show that the connectivity of $D(P)$ is at least $binom{lfloorfrac{n-2}{2}rfloor}{2}+binom{lceilfrac{n-2}{2}rceil}{2}$, and that this bound is tight for each $ngeq 3$.
{"title":"On the connectivity of the disjointness graph of segments of point sets in general position in the plane","authors":"J. Leaños, M. K. C. Ndjatchi, L. M. R'ios-Castro","doi":"10.46298/dmtcs.6678","DOIUrl":"https://doi.org/10.46298/dmtcs.6678","url":null,"abstract":"Let $P$ be a set of $ngeq 3$ points in general position in the plane. The\u0000edge disjointness graph $D(P)$ of $P$ is the graph whose vertices are all the\u0000closed straight line segments with endpoints in $P$, two of which are adjacent\u0000in $D(P)$ if and only if they are disjoint. We show that the connectivity of\u0000$D(P)$ is at least\u0000$binom{lfloorfrac{n-2}{2}rfloor}{2}+binom{lceilfrac{n-2}{2}rceil}{2}$,\u0000and that this bound is tight for each $ngeq 3$.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131641076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
O. Bodini, M. Dien, Antoine Genitrini, F. Peschanski
International audience In this paper we address the problem of understanding Concurrency Theory from a combinatorial point of view. We are interested in quantitative results and algorithmic tools to refine our understanding of the classical combinatorial explosion phenomenon arising in concurrency. This paper is essentially focusing on the the notion of synchronization from the point of view of combinatorics. As a first step, we address the quantitative problem of counting the number of executions of simple processes interacting with synchronization barriers. We elaborate a systematic decomposition of processes that produces a symbolic integral formula to solve the problem. Based on this procedure, we develop a generic algorithm to generate process executions uniformly at random. For some interesting sub-classes of processes we propose very efficient counting and random sampling algorithms. All these algorithms have one important characteristic in common: they work on the control graph of processes and thus do not require the explicit construction of the state-space.
{"title":"Quantitative and Algorithmic aspects of Barrier Synchronization in Concurrency","authors":"O. Bodini, M. Dien, Antoine Genitrini, F. Peschanski","doi":"10.46298/dmtcs.5820","DOIUrl":"https://doi.org/10.46298/dmtcs.5820","url":null,"abstract":"International audience\u0000 \u0000 In this paper we address the problem of understanding Concurrency Theory from a combinatorial point of view. We are interested in quantitative results and algorithmic tools to refine our understanding of the classical combinatorial explosion phenomenon arising in concurrency. This paper is essentially focusing on the the notion of synchronization from the point of view of combinatorics. As a first step, we address the quantitative problem of counting the number of executions of simple processes interacting with synchronization barriers. We elaborate a systematic decomposition of processes that produces a symbolic integral formula to solve the problem. Based on this procedure, we develop a generic algorithm to generate process executions uniformly at random. For some interesting sub-classes of processes we propose very efficient counting and random sampling algorithms. All these algorithms have one important characteristic in common: they work on the control graph of processes and thus do not require the explicit construction of the state-space.\u0000","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"227 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116385583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recently, a conjecture due to Hendry was disproved which stated that every Hamiltonian chordal graph is cycle extendible. Here we further explore the conjecture, showing that it fails to hold even when a number of extra conditions are imposed. In particular, we show that Hendry's Conjecture fails for strongly chordal graphs, graphs with high connectivity, and if we relax the definition of "cycle extendible" considerably. We also consider the original conjecture from a subtree intersection model point of view, showing that a result of Abuieda et al is nearly best possible.
{"title":"Further results on Hendry's Conjecture","authors":"Manuel Lafond, Ben Seamone, R. Sherkati","doi":"10.46298/dmtcs.6700","DOIUrl":"https://doi.org/10.46298/dmtcs.6700","url":null,"abstract":"Recently, a conjecture due to Hendry was disproved which stated that every\u0000Hamiltonian chordal graph is cycle extendible. Here we further explore the\u0000conjecture, showing that it fails to hold even when a number of extra\u0000conditions are imposed. In particular, we show that Hendry's Conjecture fails\u0000for strongly chordal graphs, graphs with high connectivity, and if we relax the\u0000definition of \"cycle extendible\" considerably. We also consider the original\u0000conjecture from a subtree intersection model point of view, showing that a\u0000result of Abuieda et al is nearly best possible.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133693332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a non-monotone activation process $(X_t)_{tin{ 0,1,2,ldots}}$ on a graph $G$, where $X_0subseteq V(G)$, $X_t={ uin V(G):|N_G(u)cap X_{t-1}|geq tau(u)}$ for every positive integer $t$, and $tau:V(G)to mathbb{Z}$ is a threshold function. The set $X_0$ is a so-called non-monotone target set for $(G,tau)$ if there is some $t_0$ such that $X_t=V(G)$ for every $tgeq t_0$. Ben-Zwi, Hermelin, Lokshtanov, and Newman [Discrete Optimization 8 (2011) 87-96] asked whether a target set of minimum order can be determined efficiently if $G$ is a tree. We answer their question in the affirmative for threshold functions $tau$ satisfying $tau(u)in { 0,1,d_G(u)}$ for every vertex~$u$. For such restricted threshold functions, we give a characterization of target sets that allows to show that the minimum target set problem remains NP-hard for planar graphs of maximum degree $3$ but is efficiently solvable for graphs of bounded treewidth.
{"title":"Non-monotone target sets for threshold values restricted to 0, 1, and the vertex degree","authors":"Julien Baste, S. Ehard, D. Rautenbach","doi":"10.46298/dmtcs.6844","DOIUrl":"https://doi.org/10.46298/dmtcs.6844","url":null,"abstract":"We consider a non-monotone activation process $(X_t)_{tin{ 0,1,2,ldots}}$\u0000on a graph $G$, where $X_0subseteq V(G)$, $X_t={ uin V(G):|N_G(u)cap\u0000X_{t-1}|geq tau(u)}$ for every positive integer $t$, and $tau:V(G)to\u0000mathbb{Z}$ is a threshold function. The set $X_0$ is a so-called non-monotone\u0000target set for $(G,tau)$ if there is some $t_0$ such that $X_t=V(G)$ for every\u0000$tgeq t_0$. Ben-Zwi, Hermelin, Lokshtanov, and Newman [Discrete Optimization 8\u0000(2011) 87-96] asked whether a target set of minimum order can be determined\u0000efficiently if $G$ is a tree. We answer their question in the affirmative for\u0000threshold functions $tau$ satisfying $tau(u)in { 0,1,d_G(u)}$ for every\u0000vertex~$u$. For such restricted threshold functions, we give a characterization\u0000of target sets that allows to show that the minimum target set problem remains\u0000NP-hard for planar graphs of maximum degree $3$ but is efficiently solvable for\u0000graphs of bounded treewidth.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"104 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122631425","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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