On the algorithmic aspects of strong subcoloring

A partition of the vertex set V(G) of a graph G into \(V(G)=V_1\cup V_2\cup \cdots \cup V_k\) is called a k-strong subcoloring if \(d(x,y)\ne 2\) in G for every \(x,y\in V_i\), \(1\le i \le k\) where d(x, y) denotes the length of a shortest x-y path in G. The strong subchromatic number is defined as \(\chi _{sc}(G)=\text {min}\{ k:G \text { admits a }k\)-\(\text {strong subcoloring}\}\). In this paper, we explore the complexity status of the StrongSubcoloring problem: for a given graph G and a positive integer k, StrongSubcoloring is to decide whether G admits a k-strong subcoloring. We prove that StrongSubcoloring is NP-complete for subcubic bipartite graphs and the problem is polynomial time solvable for trees. In addition, we prove the following dichotomy results: (i) for the class of \(K_{1,r}\)-free split graphs, StrongSubcoloring is in P when \(r\le 3\) and NP-complete when \(r>3\) and (ii) for the class of H-free graphs, StrongSubcoloring is polynomial time solvable only if H is an induced subgraph of \(P_4\); otherwise the problem is NP-complete. Next, we consider a lower bound on the strong subchromatic number. A strong set is a set S of vertices of a graph G such that for every \(x,y\in S\), \(d(x,y)= 2\) in G and the cardinality of a maximum strong set in G is denoted by \(\alpha _{s}(G)\). Clearly, \(\alpha _{s}(G)\le \chi _{sc}(G)\). We consider the complexity status of the StrongSet problem: given a graph G and a positive integer k, StrongSet asks whether G contains a strong set of cardinality k. We prove that StrongSet is NP-complete for (i) bipartite graphs and (ii) \(K_{1,4}\)-free split graphs, and it is polynomial time solvable for (i) trees and (ii) \(P_4\)-free graphs.     

Anti-forcing spectra of perfect matchings of graphs

Let M be a perfect matching of a graph G. The smallest number of edges whose removal to make M as the unique perfect matching in the resulting subgraph is called the anti-forcing number of M. The anti-forcing spectrum of G is the set of anti-forcing numbers of all perfect matchings in G, denoted by \(\hbox {Spec}_{af}(G)\). In this paper, we show that any finite set of positive integers can be the anti-forcing spectrum of a graph. We present two classes of hexagonal systems whose anti-forcing spectra are integer intervals. Finally, we show that determining the anti-forcing number of a perfect matching of a bipartite graph with maximum degree four is a NP-complete problem.     

On computing a minimum secure dominating set in block graphs

In a graph \(G=(V,E)\), a set \(D \subseteq V\) is said to be a dominating set of G if for every vertex \(u\in V{\setminus }D\), there exists a vertex \(v\in D\) such that \(uv\in E\). A secure dominating set of the graph G is a dominating set D of G such that for every \(u\in V{\setminus }D\), there exists a vertex \(v\in D\) such that \(uv\in E\) and \((D{\setminus }\{v\})\cup \{u\}\) is a dominating set of G. Given a graph G and a positive integer k, the secure domination problem is to decide whether G has a secure dominating set of cardinality at most k. The secure domination problem has been shown to be NP-complete for chordal graphs via split graphs and for bipartite graphs. In Liu et al. (in: Proceedings of 27th workshop on combinatorial mathematics and computation theory, 2010), it is asked to find a polynomial time algorithm for computing a minimum secure dominating set in a block graph. In this paper, we answer this by presenting a linear time algorithm to compute a minimum secure dominating set in block graphs. We then strengthen the known NP-completeness of the secure domination problem by showing that the secure domination problem is NP-complete for undirected path graphs and chordal bipartite graphs.     

Complexity properties of complementary prisms

The complementary prism \(G\bar{G}\) of a graph G arises from the disjoint union of the graph G and its complement \(\bar{G}\) by adding the edges of a perfect matching joining pairs of corresponding vertices of G and \(\bar{G}\). Haynes, Henning, Slater, and van der Merwe introduced the complementary prism and as a variation of the well-known prism. We study algorithmic/complexity properties of complementary prisms with respect to cliques, independent sets, k-domination, and especially \(P_3\)-convexity. We establish hardness results and identify some efficiently solvable cases.     

A new graph parameter and a construction of larger graph without increasing radio <Emphasis Type="Italic">k</Emphasis>-chromatic number

For a positive integer \(k\ge 2\), the radio k-coloring problem is an assignment L of non-negative integers (colors) to the vertices of a finite simple graph G satisfying the condition \(|L(u)-L(v)| \ge k+1-d(u,v)\), for any two distinct vertices u, v of G and d(u, v) being distance between u, v. The span of L is the largest integer assigned by L, while 0 is taken as the smallest color. An \(rc_k\)-coloring on G is a radio k-coloring on G of minimum span which is referred as the radio k-chromatic number of G and denoted by \(rc_k(G)\). An integer h, \(0<h<rc_k(G)\), is a hole in a \(rc_k\)-coloring on G if h is not assigned by it. In this paper, we construct a larger graph from a graph of a certain class by using a combinatorial property associated with \((k-1)\) consecutive holes in any \(rc_k\)-coloring of a graph. Exploiting the same property, we introduce a new graph parameter, referred as \((k-1)\)-hole index of G and denoted by \(\rho _k(G)\). We also explore several properties of \(\rho _k(G)\) including its upper bound and relation with the path covering number of the complement \(G^c\).     

A linear-time algorithm for clique-coloring problem in circular-arc graphs

A maximal clique of G is a clique not properly contained in any other clique. A k-clique-coloring of a graph G is an assignment of k colors to the vertices of G such that no maximal clique with at least two vertices is monochromatic. The smallest integer k admitting a k-clique-coloring of G is called clique-coloring number of G. Cerioli and Korenchendler (Electron Notes Discret Math 35:287–292, 2009) showed that there is a polynomial-time algorithm to solve the clique-coloring problem in circular-arc graphs and asked whether there exists a linear-time algorithm to find an optimal clique-coloring in circular-arc graphs or not. In this paper we present a linear-time algorithm of the optimal clique-coloring in circular-arc graphs.     

Minimum choosability of planar graphs

The problem of list coloring of graphs appears in practical problems concerning channel or frequency assignment. In this paper, we study the minimum number of choosability of planar graphs. A graph G is edge-k-choosable if whenever every edge x is assigned with a list of at least k colors, L(x)), there exists an edge coloring \(\phi \) such that \(\phi (x) \in L(x)\). Similarly, A graph G is toal-k-choosable if when every element (edge or vertex) x is assigned with a list of at least k colors, L(x), there exists an total coloring \(\phi \) such that \(\phi (x) \in L(x)\). We proved \(\chi '_{l}(G)=\Delta \) and \(\chi ''_{l}(G)=\Delta +1\) for a planar graph G with maximum degree \(\Delta \ge 8\) and without chordal 6-cycles, where the list edge chromatic number \(\chi '_{l}(G)\) of G is the smallest integer k such that G is edge-k-choosable and the list total chromatic number \(\chi ''_{l}(G)\) of G is the smallest integer k such that G is total-k-choosable.     

On the vertex cover $$P_3$$ problem parameterized by treewidth

Consider a graph G. A subset of vertices, F, is called a vertex cover \(P_t\) (\(VCP_t\)) set if every path of order t contains at least one vertex in F. Finding a minimum \(VCP_t\) set in a graph is is NP-hard for any integer \(t\ge 2\) and is called the \(MVCP_3\) problem. In this paper, we study the parameterized algorithms for the \(MVCP_3\) problem when the underlying graph G is parameterized by the treewidth. Given an n-vertex graph together with its tree decomposition of width at most p, we present an algorithm running in time \(4^{p}\cdot n^{O(1)}\) for the \(MVCP_3\) problem. Moreover, we show that for the \(MVCP_3\) problem on planar graphs, there is a subexponential parameterized algorithm running in time \(2^{O(\sqrt{k})}\cdot n^{O(1)}\) where k is the size of the optimal solution.     

Is there any polynomial upper bound for the universal labeling of graphs?

A universal labeling of a graph G is a labeling of the edge set in G such that in every orientation \(\ell \) of G for every two adjacent vertices v and u, the sum of incoming edges of v and u in the oriented graph are different from each other. The universal labeling number of a graph G is the minimum number k such that G has universal labeling from \(\{1,2,\ldots , k\}\) denoted it by \(\overrightarrow{\chi _{u}}(G) \). We have \(2\Delta (G)-2 \le \overrightarrow{\chi _{u}} (G)\le 2^{\Delta (G)}\), where \(\Delta (G)\) denotes the maximum degree of G. In this work, we offer a provocative question that is: “Is there any polynomial function f such that for every graph G, \(\overrightarrow{\chi _{u}} (G)\le f(\Delta (G))\)?”. Towards this question, we introduce some lower and upper bounds on their parameter of interest. Also, we prove that for every tree T, \(\overrightarrow{\chi _{u}}(T)={\mathcal {O}}(\Delta ^3) \). Next, we show that for a given 3-regular graph G, the universal labeling number of G is 4 if and only if G belongs to Class 1. Therefore, for a given 3-regular graph G, it is an \( {{\mathbf {N}}}{{\mathbf {P}}} \)-complete to determine whether the universal labeling number of G is 4. Finally, using probabilistic methods, we almost confirm a weaker version of the problem.     

Fractional matching preclusion of graphs

Let F be an edge subset and \(F^{\prime }\) a subset of edges and vertices of a graph G. If \(G-F\) and \(G-F^{\prime }\) have no fractional perfect matchings, then F is a fractional matching preclusion (FMP) set and \(F^{\prime }\) is a fractional strong MP (FSMP) set of G. The FMP (FSMP) number of G is the minimum size of FMP (FSMP) sets of G. In this paper, the FMP number and the FSMP number of Petersen graph, complete graphs and twisted cubes are obtained, respectively.     

Distance domination in graphs with given minimum and maximum degree

For an integer \(k \ge 1\), a distance k-dominating set of a connected graph G is a set S of vertices of G such that every vertex of V(G) is at distance at most k from some vertex of S. The distance k-domination number \(\gamma _k(G)\) of G is the minimum cardinality of a distance k-dominating set of G. In this paper, we establish an upper bound on the distance k-domination number of a graph in terms of its order, minimum degree and maximum degree. We prove that for \(k \ge 2\), if G is a connected graph with minimum degree \(\delta \ge 2\) and maximum degree \(\Delta \) and of order \(n \ge \Delta + k - 1\), then \(\gamma _k(G) \le \frac{n + \delta - \Delta }{\delta + k - 1}\). This result improves existing known results.     

The <Emphasis Type="Italic">k</Emphasis>-hop connected dominating set problem: approximation and hardness

Let G be a connected graph and k be a positive integer. A vertex subset D of G is a k-hop connected dominating set if the subgraph of G induced by D is connected, and for every vertex v in G there is a vertex u in D such that the distance between v and u in G is at most k. We study the problem of finding a minimum k-hop connected dominating set of a graph (\({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\)). We prove that \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\) is \(\mathscr {NP}\)-hard on planar bipartite graphs of maximum degree 4. We also prove that \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\) is \(\mathscr {APX}\)-complete on bipartite graphs of maximum degree 4. We present inapproximability thresholds for \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\) on bipartite and on (1, 2)-split graphs. Interestingly, one of these thresholds is a parameter of the input graph which is not a function of its number of vertices. We also discuss the complexity of computing this graph parameter. On the positive side, we show an approximation algorithm for \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\). Finally, when \(k=1\), we present two new approximation algorithms for the weighted version of the problem restricted to graphs with a polynomially bounded number of minimal separators.     

A simpler PTAS for connected <Emphasis Type="Italic">k</Emphasis>-path vertex cover in homogeneous wireless sensor network

Because of its application in the field of security in wireless sensor networks, k-path vertex cover (\(\hbox {VCP}_k\)) has received a lot of attention in recent years. Given a graph \(G=(V,E)\), a vertex set \(C\subseteq V\) is a k-path vertex cover (\(\hbox {VCP}_k\)) of G if every path on k vertices has at least one vertex in C, and C is a connected k-path vertex cover of G (\(\hbox {CVCP}_k\)) if furthermore the subgraph of G induced by C is connected. A homogeneous wireless sensor network can be modeled as a unit disk graph. This paper presents a new PTAS for \(\hbox {MinCVCP}_k\) on unit disk graphs. Compared with previous PTAS given by Liu et al., our method not only simplifies the algorithm and reduces the time-complexity, but also simplifies the analysis by a large amount.     

On the coefficients of the independence polynomial of graphs

An independent set of a graph G is a set of pairwise non-adjacent vertices. Let \(i_k = i_k(G)\) be the number of independent sets of cardinality k of G. The independence polynomial \(I(G, x)=\sum _{k\geqslant 0}i_k(G)x^k\) defined first by Gutman and Harary has been the focus of considerable research recently, whereas \(i(G)=I(G, 1)\) is called the Merrifield–Simmons index of G. In this paper, we first proved that among all trees of order n,  the kth coefficient \(i_k\) is smallest when the tree is a path, and is largest for star. Moreover, the graph among all trees of order n with diameter at least d whose all coefficients of I(G, x) are largest is identified. Then we identify the graphs among the n-vertex unicyclic graphs (resp. n-vertex connected graphs with clique number \(\omega \)) which simultaneously minimize all coefficients of I(G, x), whereas the opposite problems of simultaneously maximizing all coefficients of I(G, x) among these two classes of graphs are also solved respectively. At last we characterize the graph among all the n-vertex connected graph with chromatic number \(\chi \) (resp. vertex connectivity \(\kappa \)) which simultaneously minimize all coefficients of I(G, x). Our results may deduce some known results on Merrifield–Simmons index of graphs.     

The adjacent vertex distinguishing total chromatic numbers of planar graphs with $$\Delta =10$$

A (proper) total-k-coloring of a graph G is a mapping \(\phi : V (G) \cup E(G)\mapsto \{1, 2, \ldots , k\}\) such that any two adjacent elements in \(V (G) \cup E(G)\) receive different colors. Let C(v) denote the set of the color of a vertex v and the colors of all incident edges of v. A total-k-adjacent vertex distinguishing-coloring of G is a total-k-coloring of G such that for each edge \(uv\in E(G)\), \(C(u)\ne C(v)\). We denote the smallest value k in such a coloring of G by \(\chi ''_{a}(G)\). It is known that \(\chi _{a}''(G)\le \Delta (G)+3\) for any planar graph with \(\Delta (G)\ge 11\). In this paper, we show that if G is a planar graph with \(\Delta (G)\ge 10\), then \(\chi _{a}''(G)\le \Delta (G)+3\). Our approach is based on Combinatorial Nullstellensatz and the discharging method.     

On the complete width and edge clique cover problems

A complete graph is the graph in which every two vertices are adjacent. For a graph \(G=(V,E)\), the complete width of G is the minimum k such that there exist k independent sets \(\mathtt {N}_i\subseteq V\), \(1\le i\le k\), such that the graph \(G'\) obtained from G by adding some new edges between certain vertices inside the sets \(\mathtt {N}_i\), \(1\le i\le k\), is a complete graph. The complete width problem is to decide whether the complete width of a given graph is at most k or not. In this paper we study the complete width problem. We show that the complete width problem is NP-complete on \(3K_2\)-free bipartite graphs and polynomially solvable on \(2K_2\)-free bipartite graphs and on \((2K_2,C_4)\)-free graphs. As a by-product, we obtain the following new results: the edge clique cover problem is NP-complete on \(\overline{3K_2}\)-free co-bipartite graphs and polynomially solvable on \(C_4\)-free co-bipartite graphs and on \((2K_2, C_4)\)-free graphs. We also give a characterization for k-probe complete graphs which implies that the complete width problem admits a kernel of at most \(2^k\) vertices. This provides another proof for the known fact that the edge clique cover problem admits a kernel of at most \(2^k\) vertices. Finally we determine all graphs of small complete width \(k\le 3\).     

An $$O(|E(G)|^2)$$ algorithm for recognizing Pfaffian graphs of a type of bipartite graphs

A graph \(G=(V,E)\) with even number vertices is called Pfaffian if it has a Pfaffian orientation, namely it admits an orientation such that the number of edges of any M-alternating cycle which have the same direction as the traversal direction is odd for some perfect matching M of the graph G. In this paper, we obtain a necessary and sufficient condition of Pfaffian graphs in a type of bipartite graphs. Then, we design an \(O(|E(G)|^2)\) algorithm for recognizing Pfaffian graphs in this class and constructs a Pfaffian orientation if the graph is Pfaffian. The results improve and generalize some known results.     

Neighbor-sum-distinguishing edge choosability of subcubic graphs

A graph G is said to be neighbor-sum-distinguishing edge k-choose if, for every list L of colors such that L(e) is a set of k positive real numbers for every edge e, there exists a proper edge coloring which assigns to each edge a color from its list so that for each pair of adjacent vertices u and v the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. Let \(\mathrm{ch}^{\prime }_{\sum ^p}(G)\) denote the smallest integer k such that G is neighbor-sum-distinguishing edge k-choose. In this paper, we prove that if G is a subcubic graph with the maximum average degree mad(G), then (1) \(\mathrm{ch}^{\prime }_{\sum ^p}(G)\le 7\); (2) \(\mathrm{ch}^{\prime }_{\sum ^p}(G)\le 6\) if \(\hbox {mad}(G)<\frac{36}{13}\); (3) \(\mathrm{ch}^{\prime }_{\sum ^p}(G)\le 5\) if \(\hbox {mad}(G)<\frac{5}{2}\).     

List-edge-coloring of planar graphs without 6-cycles with three chords

A graph G is edge-k-choosable if, whenever we are given a list L(e) of colors with \(|L(e)|\ge k\) for each \(e\in E(G)\), we can choose a color from L(e) for each edge e such that no two adjacent edges receive the same color. In this paper we prove that if G is a planar graph, and each 6-cycle contains at most two chords, then G is edge-k-choosable, where \(k=\max \{8,\Delta (G)+1\}\), and edge-t-choosable, where \(t=\max \{10,\Delta (G)\}\).     

Paired-domination number of claw-free odd-regular graphs

A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching, while the paired-domination number is the minimum cardinality of a paired-dominating set in the graph, denoted by \(\gamma _{pr}(G)\). Let G be a connected \(\{K_{1,3}, K_{4}-e\}\)-free cubic graph of order n. We show that \(\gamma _{pr}(G)\le \frac{10n+6}{27}\) if G is \(C_{4}\)-free and that \(\gamma _{pr}(G)\le \frac{n}{3}+\frac{n+6}{9(\lceil \frac{3}{4}(g_o+1)\rceil +1)}\) if G is \(\{C_{4}, C_{6}, C_{10}, \ldots , C_{2g_o}\}\)-free for an odd integer \(g_o\ge 3\); the extremal graphs are characterized; we also show that if G is a 2 -connected, \(\gamma _{pr}(G) = \frac{n}{3} \). Furthermore, if G is a connected \((2k+1)\)-regular \(\{K_{1,3}, K_4-e\}\)-free graph of order n, then \(\gamma _{pr}(G)\le \frac{n}{k+1} \), with equality if and only if \(G=L(F)\), where \(F\cong K_{1, 2k+2}\), or k is even and \(F\cong K_{k+1,k+2}\).