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 (vertex-)monochromatic index of a graph

A tree T in an edge-colored (vertex-colored) graph H is called a monochromatic (vertex-monochromatic) tree if all the edges (internal vertices) of T have the same color. For \(S\subseteq V(H)\), a monochromatic (vertex-monochromatic) S-tree in H is a monochromatic (vertex-monochromatic) tree of H containing the vertices of S. For a connected graph G and a given integer k with \(2\le k\le |V(G)|\), the k -monochromatic index \(mx_k(G)\) (k -monochromatic vertex-index \(mvx_k(G)\)) of G is the maximum number of colors needed such that for each subset \(S\subseteq V(G)\) of k vertices, there exists a monochromatic (vertex-monochromatic) S-tree. For \(k=2\), Caro and Yuster showed that \(mc(G)=mx_2(G)=|E(G)|-|V(G)|+2\) for many graphs, but it is not true in general. In this paper, we show that for \(k\ge 3\), \(mx_k(G)=|E(G)|-|V(G)|+2\) holds for any connected graph G, completely determining the value. However, for the vertex-version \(mvx_k(G)\) things will change tremendously. We show that for a given connected graph G, and a positive integer L with \(L\le |V(G)|\), to decide whether \(mvx_k(G)\ge L\) is NP-complete for each integer k such that \(2\le k\le |V(G)|\). Finally, we obtain some Nordhaus–Gaddum-type results for the k-monochromatic vertex-index.     

Algorithm complexity of neighborhood total domination and $$(\rho ,\gamma _{nt})$$-graphs

A neighborhood total dominating set, abbreviated for NTD-set D, is a vertex set of G such that D is a dominating set with an extra property: the subgraph induced by the open neighborhood of D has no isolated vertex. The neighborhood total domination number, denoted by \(\gamma _{nt}(G)\), is the minimum cardinality of a NTD-set in G. In this paper, we prove that NTD problem is NP-complete for bipartite graphs and split graphs. Then we give a linear-time algorithm to determine \(\gamma _{nt}(T)\) for a given tree T. Finally, we characterize a constructive property of \((\gamma _{nt},2\gamma )\)-trees and provide a constructive characterization for \((\rho ,\gamma _{nt})\)-graphs, where \(\gamma \) and \(\rho \) are domination number and packing number for the given graph, respectively.     

On maximum Wiener index of trees and graphs with given radius

Let G be a connected graph of order n. The long-standing open and close problems in distance graph theory are: what is the Wiener index W(G) or average distance \(\mu (G)\) among all graphs of order n with diameter d (radius r)? There are very few number of articles where were worked on the relationship between radius or diameter and Wiener index. In this paper, we give an upper bound on Wiener index of trees and graphs in terms of number of vertices n, radius r, and characterize the extremal graphs. Moreover, from this result we give an upper bound on \(\mu (G)\) in terms of order and independence number of graph G. Also we present another upper bound on Wiener index of graphs in terms of number of vertices n, radius r and maximum degree \(\Delta \), and characterize the extremal graphs.     

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.     

Coupon coloring of some special graphs

Let G be a graph without isolated vertices. A k-coupon coloring of G is a k-coloring of G such that the neighborhood of every vertex of G contains vertices of all colors from \([k] =\{1, 2, \ldots , k\}\), which was recently introduced by Chen, Kim, Tait and Verstraete. The coupon coloring number \(\chi _c(G)\) of G is the maximum k for which a k-coupon coloring exists. In this paper, we mainly study the coupon coloring of some special classes of graphs. We determine the coupon coloring numbers of complete graphs, complete k-partite graphs, wheels, cycles, unicyclic graphs, bicyclic graphs and generalised \(\Theta \)-graphs.     

A characterization of linearizable instances of the quadratic minimum spanning tree problem

We investigate special cases of the quadratic minimum spanning tree problem (QMSTP) on a graph \(G=(V,E)\) that can be solved as a linear minimum spanning tree problem. We give a characterization of such problems when G is a complete graph, which is the standard case in the QMSTP literature. We extend our characterization to a larger class of graphs that include complete bipartite graphs and cactuses, among others. Our characterization can be verified in \(O(|E|^2)\) time. In the case of complete graphs and when the cost matrix is given in factored form, we show that our characterization can be verified in O(|E|) time. Related open problems are also indicated.     

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 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.     

Embeddings into almost self-centered graphs of given radius

A graph is almost self-centered (ASC) if all but two of its vertices are central. An almost self-centered graph with radius r is called an r-ASC graph. The r-ASC index \(\theta _r(G)\) of a graph G is the minimum number of vertices needed to be added to G such that an r-ASC graph is obtained that contains G as an induced subgraph. It is proved that \(\theta _r(G)\le 2r\) holds for any graph G and any \(r\ge 2\) which improves the earlier known bound \(\theta _r(G)\le 2r+1\). It is further proved that \(\theta _r(G)\le 2r-1\) holds if \(r\ge 3\) and G is of order at least 2. The 3-ASC index of complete graphs is determined. It is proved that \(\theta _3(G)\in \{3,4\}\) if G has diameter 2 and for several classes of graphs of diameter 2 the exact value of the 3-ASC index is obtained. For instance, if a graph G of diameter 2 does not contain a diametrical triple, then \(\theta _3(G) = 4\). The 3-ASC index of paths of order \(n\ge 1\), cycles of order \(n\ge 3\), and trees of order \(n\ge 10\) and diameter \(n-2\) are also determined, respectively, and several open problems proposed.     

Neighbor product distinguishing total colorings

A total-[k]-coloring of a graph G is a mapping \(\phi : V (G) \cup E(G)\rightarrow \{1, 2, \ldots , k\}\) such that any two adjacent elements in \(V (G) \cup E(G)\) receive different colors. Let f(v) denote the product of the color of a vertex v and the colors of all edges incident to v. A total-[k]-neighbor product distinguishing-coloring of G is a total-[k]-coloring of G such that \(f(u)\ne f(v)\), where \(uv\in E(G)\). By \(\chi ^{\prime \prime }_{\prod }(G)\), we denote the smallest value k in such a coloring of G. We conjecture that \(\chi _{\prod }^{\prime \prime }(G)\le \Delta (G)+3\) for any simple graph with maximum degree \(\Delta (G)\). In this paper, we prove that the conjecture holds for complete graphs, cycles, trees, bipartite graphs and subcubic graphs. Furthermore, we show that if G is a \(K_4\)-minor free graph with \(\Delta (G)\ge 4\), then \(\chi _{\prod }^{\prime \prime }(G)\le \Delta (G)+2\).     

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.     

Total and forcing total edge-to-vertex monophonic number of a graph

For a connected graph \(G = \left( V,E\right) \), a set \(S\subseteq E(G)\) is called a total edge-to-vertex monophonic set of a connected graph G if the subgraph induced by S has no isolated edges. The total edge-to-vertex monophonic number \(m_{tev}(G)\) of G is the minimum cardinality of its total edge-to-vertex monophonic set of G. The total edge-to-vertex monophonic number of certain classes of graphs is determined and some of its general properties are studied. Connected graphs of size \(q \ge 3 \) with total edge-to-vertex monophonic number q is characterized. It is shown that for positive integers \(r_{m},d_{m}\) and \(l\ge 4\) with \(r_{m}< d_{m} \le 2 r_{m}\), there exists a connected graph G with \(\textit{rad}_ {m} G = r_{m}\), \(\textit{diam}_ {m} G = d_{m}\) and \(m_{tev}(G) = l\) and also shown that for every integers a and b with \(2 \le a \le b\), there exists a connected graph G such that \( m_{ev}\left( G\right) = b\) and \(m_{tev}(G) = a + b\). A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing total edge-to-vertex monophonic number of S, denoted by \(f_{tev}(S)\) is the cardinality of a minimum forcing subset of S. The forcing total edge-to-vertex monophonic number of G, denoted by \(f_{tev}(G) = \textit{min}\{f_{tev}(S)\}\), where the minimum is taken over all total edge-to-vertex monophonic set S in G. The forcing total edge-to-vertex monophonic number of certain classes of graphs are determined and some of its general properties are studied. It is shown that for every integers a and b with \(0 \le a \le b\) and \(b \ge 2\), there exists a connected graph G such that \(f_{tev}(G) = a\) and \( m _{tev}(G) = b\), where \( f _{tev}(G)\) is the forcing total edge-to-vertex monophonic number of G.     

Total coloring of 1-toroidal graphs with maximum degree at least 11 and no adjacent triangles

A total coloring of a graph G is an assignment of colors to the vertices and the edges of G such that every pair of adjacent/incident elements receive distinct colors. The total chromatic number of a graph G, denoted by \(\chi ''(G)\), is the minimum number of colors in a total coloring of G. The well-known total coloring conjecture (TCC) says that every graph with maximum degree \(\Delta \) admits a total coloring with at most \(\Delta + 2\) colors. A graph is 1-toroidal if it can be drawn in torus such that every edge crosses at most one other edge. In this paper, we investigate the total coloring of 1-toroidal graphs, and prove that the TCC holds for the 1-toroidal graphs with maximum degree at least 11 and some restrictions on the triangles. Consequently, if G is a 1-toroidal graph with maximum degree \(\Delta \) at least 11 and without adjacent triangles, then G admits a total coloring with at most \(\Delta + 2\) colors.     

The $$(k,\ell )$$-proper index of graphs

A tree T in an edge-colored graph is called a proper tree if no two adjacent edges of T receive the same color. Let G be a connected graph of order n and k be an integer with \(2\le k \le n\). For \(S\subseteq V(G)\) and \(|S| \ge 2\), an S-tree is a tree containing the vertices of S in G. A set \(\{T_1,T_2,\ldots ,T_\ell \}\) of S-trees is called internally disjoint if \(E(T_i)\cap E(T_j)=\emptyset \) and \(V(T_i)\cap V(T_j)=S\) for \(1\le i\ne j\le \ell \). For a set S of k vertices of G, the maximum number of internally disjoint S-trees in G is denoted by \(\kappa (S)\). The k-connectivity \(\kappa _k(G)\) of G is defined by \(\kappa _k(G)=\min \{\kappa (S)\mid S\) is a k-subset of \(V(G)\}\). For a connected graph G of order n and for two integers k and \(\ell \) with \(2\le k\le n\) and \(1\le \ell \le \kappa _k(G)\), the \((k,\ell )\)-proper index \(px_{k,\ell }(G)\) of G is the minimum number of colors that are required in an edge-coloring of G such that for every k-subset S of V(G), there exist \(\ell \) internally disjoint proper S-trees connecting them. In this paper, we show that for every pair of positive integers k and \(\ell \) with \(k \ge 3\) and \(\ell \le \kappa _k(K_{n,n})\), there exists a positive integer \(N_1=N_1(k,\ell )\) such that \(px_{k,\ell }(K_n) = 2\) for every integer \(n \ge N_1\), and there exists also a positive integer \(N_2=N_2(k,\ell )\) such that \(px_{k,\ell }(K_{m,n}) = 2\) for every integer \(n \ge N_2\) and \(m=O(n^r) (r \ge 1)\). In addition, we show that for every \(p \ge c\root k \of {\frac{\log _a n}{n}}\) (\(c \ge 5\)), \(px_{k,\ell }(G_{n,p})\le 2\) holds almost surely, where \(G_{n,p}\) is the Erd?s–Rényi random graph model.     

On the b-coloring of tight graphs

A coloring c of a graph \(G=(V,E)\) is a b -coloring if for every color i there is a vertex, say w(i), of color i whose neighborhood intersects every other color class. The vertex w(i) is called a b-dominating vertex of color i. The b -chromatic number of a graph G, denoted by b(G), is the largest integer k such that G admits a b-coloring with k colors. Let m(G) be the largest integer m such that G has at least m vertices of degree at least \(m-1\). A graph G is tight if it has exactly m(G) vertices of degree \(m(G)-1\), and any other vertex has degree at most \(m(G)-2\). In this paper, we show that the b-chromatic number of tight graphs with girth at least 8 is at least \(m(G)-1\) and characterize the graphs G such that \(b(G)=m(G)\). Lin and Chang (2013) conjectured that the b-chromatic number of any graph in \(\mathcal {B}_{m}\) is m or \(m-1\) where \(\mathcal {B}_{m}\) is the class of tight bipartite graphs \((D,D{^\prime })\) of girth 6 such that D is the set of vertices of degree \(m-1\). We verify the conjecture of Lin and Chang for some subclass of \(\mathcal {B}_{m}\), and we give a lower bound for any graph in \(\mathcal {B}_{m}\).     

Algorithms for connected <Emphasis Type="Italic">p</Emphasis>-centdian problem on block graphs

We consider the facility location problem of locating a set \(X_p\) of p facilities (resources) on a network (or a graph) such that the subnetwork (or subgraph) induced by the selected set \(X_p\) is connected. Two problems on a block graph G are proposed: one problem is to minimizes the sum of its weighted distances from all vertices of G to \(X_p\), another problem is to minimize the maximum distance from each vertex that is not in \(X_p\) to \(X_p\) and, at the same time, to minimize the sum of its distances from all vertices of G to \(X_p\). We prove that the first problem is linearly solvable on block graphs with unit edge length. For the second problem, it is shown that the set of Pareto-optimal solutions of the two criteria has cardinality not greater than n, and can be obtained in \(O(n^2)\) time, where n is the number of vertices of the block graph G.     

Safe sets in graphs: Graph classes and structural parameters

A safe set of a graph \(G=(V,E)\) is a non-empty subset S of V such that for every component A of G[S] and every component B of \(G[V {\setminus } S]\), we have \(|A| \ge |B|\) whenever there exists an edge of G between A and B. In this paper, we show that a minimum safe set can be found in polynomial time for trees. We then further extend the result and present polynomial-time algorithms for graphs of bounded treewidth, and also for interval graphs. We also study the parameterized complexity. We show that the problem is fixed-parameter tractable when parameterized by the solution size. Furthermore, we show that this parameter lies between the tree-depth and the vertex cover number. We then conclude the paper by showing hardness for split graphs and planar graphs.     

On zero-sum $$\mathbb {Z}_{2j}^k$$-magic graphs

Let \(G = (V,E)\) be a finite graph and let \((\mathbb {A},+)\) be an abelian group with identity 0. Then G is \(\mathbb {A}\)-magic if and only if there exists a function \(\phi \) from E into \(\mathbb {A} - \{0\}\) such that for some \(c \in \mathbb {A}, \sum _{e \in E(v)} \phi (e) = c\) for every \(v \in V\), where E(v) is the set of edges incident to v. Additionally, G is zero-sum \(\mathbb {A}\)-magic if and only if \(\phi \) exists such that \(c = 0\). We consider zero-sum \(\mathbb {A}\)-magic labelings of graphs, with particular attention given to \(\mathbb {A} = \mathbb {Z}_{2j}^k\). For \(j \ge 1\), let \(\zeta _{2j}(G)\) be the smallest positive integer c such that G is zero-sum \(\mathbb {Z}_{2j}^c\)-magic if c exists; infinity otherwise. We establish upper bounds on \(\zeta _{2j}(G)\) when \(\zeta _{2j}(G)\) is finite, and show that \(\zeta _{2j}(G)\) is finite for all r-regular \(G, r \ge 2\). Appealing to classical results on the factors of cubic graphs, we prove that \(\zeta _4(G) \le 2\) for a cubic graph G, with equality if and only if G has no 1-factor. We discuss the problem of classifying cubic graphs according to the collection of finite abelian groups for which they are zero-sum group-magic.     

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 \(k\le 3\).